(The other answer is quite correct, but I wanted to provide an answer that breaks it up chronologically.)
The following is the chronological series of events, rationalizations, and experiences the characters face that reveal to them that Dr. Hannibal Lecter is the Chesapeake Ripper.
Before the beginning of the television series there is a known serial killer named The Chesapeake Ripper. He is assumed to be a white male, something foreign/"exotic" about him, but a highly skilled surgeon and quite possibly a "pure" psychopath. This is considered rare because on a spectrum of psychosis, the "pure" psychopath would be entirely aware of all their thoughts and actions, but would be entirely devoid of sympathy and empathy. Will Graham is considered to be the exact opposite, purely empathetic, and that is the source of his "special" abilities. Hannibal and Jack have a conversation about this when concerned about the effect of the work on Will's state-of-mind. But at this point, no one associated with the FBI has met the Ripper and returned to tell about them.
Also before the beginning of this series, Jack had given the fledgling agent, Miriam Lass, the opportunity to start out with the Ripper case and try to shore up some obscure leads. By investigating the victims, she found that one of them was a hunter who was admitted to the ER just before his disappearance and murder. She checked-in on who worked at the ER the night of that victim's admittance and even though he wasn't on the call sheet, she found through another ER doctor that Hannibal was there that night. He offers to give her some of his journals from that time, and while he is supposedly retrieving them Miriam notices the "The Wounded Man" sketch by Hannibal that matches one of the arrangements of the Ripper victims. At this point, he sneaks up on her and attacks her, ostensibly then killing her (as Jack Crawford always believed), but we find out that she was actually kidnapped, held hostage for 2 years, and hypnotized (and probably drugged) to induce memory loss and convince her that she was being hypnotized and held captive by Dr. Frederick Chilton. At some point in the two year span, her left arm is cut off and during season 1 it is placed in the abandoned Observatory with her cell phone as a taunt from the Ripper to Jack Crawford. This gets slightly swept around throughout the seasons that focus on Will Graham, but it indicates how the Ripper sees Jack Crawford (called "The Guru", by Miriam) as a kind of nemesis. Since Jack has been hunting the Ripper before the show even started, and that Hannibal has ostensibly been "ripping" since he was a surgeon. This makes the Chesapeake Ripper as the most sought-after serial killer by Jack Crawford's behavioural sciences unit.
Season 1 then begins with Jack enlisting Will Graham to stop a currently active abductor, assumed serial killer, who is operating in Minnesota. They aren't finding any of the victims, but they all look quite similar and are being abducted around/near college campuses. Will Graham's "special" abilities are really a combination of intellect, observation, and a natural proclivity towards "extreme" empathy. He not only observes and absorbs all the facts about the situation, and his experiences as a trained FBI profiler (though never truly a field agent, due to his autism- and psychopathy-spectrum tendencies), but he puts himself in the "shoes" of the killer and envisions their point of view. Looking at the abductions attributed to the Minnesota Shrike (Garret Jacob Hobbs), he realizes that he is kidnapping replacements for his daughter.
Elise Nichols is the Minnesota Shrike's last abduction victim, but he "returned" her because she had liver cancer, so he couldn't "honor her" by eating her, and thus decided to return her as the "honorable" thing to do. This made Garret Jacob Hobbs' psychopathy as something exactly in-between Hannibal and Will's, he has the humanity and emotionality of Will, but the impassioned cannibalistic power/feeding/god-like narcissism of Hannibal. That's why he is their constant connection. And it turns out that his daughter was being raised, slowly and somewhat reluctantly, in this lifestyle. She probably didn't want to be a part of it, and didn't share his innate psychopathy, but she was raised in it. To Will and Hannibal, she becomes something of a surrogate daughter, and again they are trying to pull her towards their extremes. She, Abigail Hobbs, heard Hannibal on the phone (with his fairly obvious accent) and realized he warned her Dad. While she doesn't know he is the Chesapeake Ripper, she is the first to realize who/what he is.
Hannibal makes it even more clear to Abigail when he allows Nicholas Boyle to confront her in her home, after Freddie Lounds pointed out that he was sneaking around the property (and Freddie had previously met and provoked him further). Freddie was trying to get "the story", but Hannibal decided to (as his theme of action) "see what would happen". Hannibal knocks out Alana and offers Abigail the choice to face what she's done, where it's clear that she killed Nicholas quickly and efficiently, like a knowing killer would, or that they can hide the body and Hannibal can guide her. She chooses to side with Hannibal, and while she still might not know who he is, she knows he's like her Dad.
Surrounding the deduction about, and death of, Garret Jacob Hobbs, there are two new murders: the body on the stag's head in the field, where the Minnesota Shrike name is (ironically) coined, and the murder of Marissa Schurr in the Hobbs' hunting cabin. The first happens before Hobbs' murder by Will, and the second happens after, but also is before Abigail is institutionalized. Jack's original theory is that Garret Jacob Hobbs killed 8 girls, but returned Elise Nichols, then killed a 9th girl (in the field), and then Will stopped him. But Will saw that the 9th girl in the field was someone else, probably the Ripper, but they call him the copycat (Hobbs' copycat). Then Marissa Schurr's murder is a collective toss-up between: Nicholas, Will, Abigail, and the Copycat. At this point, no one suspects Hannibal, except possibly Abigail, though that's never revealed. Later on we find out that Hannibal (as the Ripper) killed the 9th girl in the field as the Ripper in "tribute" to the Minnesota Shrike and to expose him to Will as he understood him but Will didn't, and then he killed Marissa Schurr to "see what would happen" and generally throw a wrench into everything that was happening. He found the rock that Marissa threw and hit Nicholas, which had a bit of his blood, and so he essentially framed Nicholas, but in a way that was obviously a frame-job from Will's point-of-view. This is where Will begins to see those two murders as being connected, perhaps to the Ripper, since he believes Abigail is all but entirely innocent.
We then learn that Dr. Chilton has accidentally manipulated (through "Psychic Driving") Dr. Abel Gideon into thinking he was the Chesapeake Ripper, and so he recreated one of the murders. However, Chilton was just trying to "make his name" as he had been consulted on the Ripper case back before the beginning of the series, and Gideon was a very malleable psychopath/sociopath, as he was especially narcissistic but unstable (unlike Hannibal's staunch self-awareness). Gideon has been imprisoned for so long, and already was unstable, so under Psychic Driving, he has no idea who he really is, and he clings to the notion that perhaps he is the Ripper, since only Will thinks that the latest murders are Ripper murders.
We also learn that Dr. Du Maurier suspects Hannibal is not all together a normal human being (wearing a Human Suit or a Veil of Humanity), but she doesn't realize what/who he really is. She starts to see, quite astutely, that he has become obsessed with Will Graham and is seeing him as some kind of person that may truly understand him. She finds it unhealthy, but he clearly ignores her, and she starts to realize that everything he's doing with the FBI may be some kind of scheme and manipulation. She isn't really sure about what happened when she killed her former patient to protect herself, but she is becoming sure that Hannibal is very dangerous and is probably manipulating her. She doesn't quite know that he's the Ripper, probably because she isn't aware of that case in detail, but she is sure that he's dangerous and detached.
Will is now "losing time" from his autoimmune form of encephalitis, where the right-half of his brain believes is inflammed from an improper, over-active immune-system response. Will's hallucinations about Garret Jacob Hobbs and author killers, all start to blend together, and he starts seeing living manifestations of the Stag statue from Dr. Lecter's office. Clearly Will has known for quite some time that Hannibal is the Ripper and performed those two murders in Minnesota, but he never really let himself believe it. So now it is slowly bubbling to the surface. He sees that Hannibal fits the profile, and when the Doctor who performed his MRI is murdered, he doesn't think it was Georgia Madchen, he thinks it was the Ripper, and why would the Ripper want to frame Georgia unless he was covering up something about Will. At this point Will knows, but he is too sick and too delusional to figure it out, so his mind is revealing it to him through the visions of the Shrike-Stag. He also is realizing that Hannibal must be a cannibal as well. While still feverish, he kidnaps the escaped Dr. Gideon and brings him to Hannibal. Will consciously thinks this is because he needs Hannibal to tell him whether his vision of Hobbs' zombie-ghost is real or delusional, but subconsciously this is Will allowing the fake Ripper to meet the real Ripper and expose everyone to each other. Hannibal takes advantage of this and Will's subsequent seizure, to then reveal himself as the Ripper to Gideon and encourage him to kill Alana Bloom, then feigning worry and encouraging Will to go after Gideon.
Will stops Gideon but also collapses, everyone is worried about him and starts to wonder if he is more than just physically ill, but also psychologically ill. Will then appears to kidnap Abigail and take her back to Minnesota, subconsciously this is because he needs to confront the truth about her role in her father's murders, but consciously he thinks he is securing their surrogate father-daughter bond. Hannibal knows about this trip, so he hides it from Jack and Alana and "sneaks" out to intercept Abigail. He fakes her murder and force-feeds Will her ear. Hannibal is revealed fully, finally, as the Ripper to Abigail, and when Will realizes he's being framed for the "copycat" murders, Nicholas Boyle's murder, and Abigail Hobbs' murder, then he knows that Hannibal did it to him.
With Will imprisoned and framed, Dr. Du Maurier realizes the extent of Hannibal's manipulations, and she realizes that he is probably the murderer. She may or may not know the extent, or the name "The Chesapeake Ripper", but she knows, and that's why she disappears before he can kill her (though Hannibal did show up in her home in his plastic murder suit).
Gideon knows Hannibal is the Ripper, but he's imprisoned again.
Freddie finds Will to be a believable murdered, so she doesn't realize what's going on.
Alana and Jack, Beverly, Price, and Zeller all believe in the evidence that Will has snapped, but they hope that it was a psychic break (a fugue state), and don't believe he's the Ripper.
Will is unrelenting in his correct belief that he was framed by Hannibal, and he is starting to get through to Jack by sowing the seeds of doubt.
Alana had trained under Hannibal as a student and was always, clearly, attracted to him, so she trusts him implicitly and even begins a romantic relationship with him.
Beverly has had a crush on Will and wants to believe him, but feels similarly betrayed, like Alana. But she accepts his push to ignore the obvious answer and just work from the evidence. She finds that the evidence leads her to Hannibal, and she realizes he's the Ripper, but she's caught and killed.
Hannibal had saved Jack's wife, through a flip of a coin, but used this as a way to secure Jack's trust in him as a friend, all the while Hannibal as the Ripper still seems him as a nemesis, taunting him with Miriam Lass.
Will gains back some memories and knows that Gideon knows Hannibal is the Ripper, so he gets Chilton to encourage his "delusions" and bring-in Gideon, so they can both say the same name, leading Chilton to the fame of having "caught" the Ripper. Chilton doesn't so much as believe Will, as he does just generally distrust everyone. Gideon is rightfully scared of the Ripper, and is still quite mentally broken, so he encourages Will to realize that the only way to really catc the Ripper is to kill him.
The nurse at the asylum is a budding psychopath and he worships the Ripper, who he thinks Will is, and so he "helped" Will by killing the Bailiff from the trial, trying to throw off new evidence suggesting that Will couldn't have done the other murders, since the same "signature" is still being used. Hannibal takes it further and kills the Judge from the trial, as the Ripper, and provides even more evidence against Will's incarceration.
Will decides to test all his options by having Beverly work the evidence angle, he uses Gideon (while Chilton obviously records them) to admit knowledge of Hannibal as the Ripper, and he encourages the nurse to go kill Hannibal. Now, though, Will is becoming more like Hannibal and less like himself.
Hannibal displays Beverley's body, in Ripper-fashion, after Jack just saved Hannibal from the Nurse and heard Chilton's recordings, so he brings out Will and now realizes that it is Hannibal, even though Hannibal saved his wife from killing herself, it's all but clear. Freddie, having found Beverley, is also starting to doubt Will's role in everything, and Gideon encouraged the guards to beat him up and he got himself put in the hospital, so he couldn't have done it either.
Hannibal then goes and kills Gideon and provides all new evidence that exonerates Will from all the murders, and reveals that every murder was "his", except the Bailiff. He even takes "credit" for Nicholas Boyle, but that's more about freeing Will than really stealing credit from Abigail. Also, he still wants to hide Abigail's true nature. Now Jack knows Will should be free, the Ripper did everything, and the Ripper is probably Hannibal. Chilton also knows that it is probably Hannibal, but is trying not to "out" himself.
Hannibal sends the bear-man to kill Will, and Will beats him to death then performs a Ripper-in-training display of the body with Hannibal's help. This made them "even" (sending someone to kill each other), and Will is showing that he's "coming around" to Hannibal's god-like detached psychopathic view of the world. Jack knows that Will can get close to Hannibal, so he's letting him. But Freddie still distrusts Will, so she snoops on him and finds the bear-man's remains in his freezer, but he temporarily kidnaps her and then brings in Jack and they reveal their plans. At this point, Alana, Price, and Zeller (and the rest of the FBI) are the only ones out of the loop. Du Maurier is gone, and Hannibal is putting his trust (blinded by his obsession and loneliness) in Will's friendship.
Ongoing also are the Verger siblings, and Hannibal is again "seeing what will happen" by encouraging Margot to kill her brother, but she runs into Will, and tricks him into impregnating her, to secure her "safety" by having an heir. She suspects Hannibal may not be quite normal, but isn't so sure he's like her brother. Hannibal is using the new child as a way to manipulate Will's emotions, to make sure they have a secure bond as surrogate fathers, reminding him of (the thought-to-be dead) Abigail. He then tells Mason about the baby, knowing he'll do something (again to "see what would happen"), and Mason has the baby aborted and has a hysterectomy performed. Will should want to kill Mason, but he knows it's Hannibal's manipulation, and he doesn't want Mason to kill Hannibal, so he frees Hannibal when they're captured, and Hannibal tortures Mason by drugging him and having him mutilate himself.
Alana, just before Mason's mutilation, was encouraged by Will to go to a gun range and be prepared to protect herself. With Will's freedom and Jack's aloofness, she is starting to question everything, including opening herself up to the idea that all the claims about Hannibal may be true. To throw everyone off his trail, Hannibal frames Chilton, revealing himself to Chilton, but leaving the FBI with no option but to believe that Chilton is the Ripper. This makes everyone who knows Chilton quite suspicious, but at this point, only Alana is out of the loop on Hannibal being the Ripper, so she takes Will's advice and goes to the gun range, which Hannibal smells on her.
The final plan, after catching Chilton (and Miriam killing/wounding him before he can tell Alana that Hannibal is the Ripper), is that Jack has Will encourage Hannibal to expose himself to Jack. Hannibal invites Jack to a final dinner, where ostensibly Hannibal and Will are planning to kill Jack and flee the country. Alana will be left behind, and everyone else will be imprisoned or dead, and Du Maurier is already gone. Everything seems to be going to plan, but Hannibal smells Freddie Lounds (who should be dead) on Will's hair and clothes. He realizes she must be alive, it must have been fake, and he must be colluding with Jack. His trust and world shattered, Hannibal decides in one desperate last meal to give Will the opportunity for them to run away together without exposing their "true natures" to Jack. This is Hannibal's hope that Will really does accept him, and could just be manipulating Jack. But Will says he "need[s] [Jack] to know". So Hannibal knows he is back to just himself.
Jack is desperate, obviously, with this plan, but knows he'll need more testimony once they catch him, so he finds and brings in Dr. Du Maurier, and finally she realizes what she probably should have known all along, that Hannibal is the Ripper, and that he is pulling all the strings. She knows that all their plans will fail, and that Hannibal is always ten steps ahead. Ostensibly (though we'll find out actually when season 3 begins) this is why she leaves with Hannibal, similar to what Freddie suspected of Will, "if you can't beat 'em, join 'em.". Since the FBI clearly couldn't stop him, she decides to join him.
At the finale, the FBI realizes what Jack was doing, so they stop him, but before they can arrest Will and Jack, Alana warns them, and then she too decides to go confront Hannibal, in case Will and Jack are on their way there, too. Jack arrives and they fight and he's hurt and trapped in the pantry when Alana arrives. She finally sees Hannibal for who he is, though she already was suspecting, but was never really in on the plan, so we can assume that she doesn't really let herself know until this moment. She runs and is confronted by Abigail who has been kidnapped by Hannibal all along. She, under Hannibal's orders, pushes Alana out the window, and Will arrives. He goes after Hannibal, but is confronted by Abigail, and is genuinely shocked and surprised. He tries to continue lying to Hannibal, because he doesn't know that Hannibal already knows everything and he worries Jack might be dead, but Hannibal cuts Will and then, as "payback" for his betrayal, he kills Abigail (just like her father first tried) all over again in front of Will, but this time Will is powerless and left for dead.
Only at this point has Alana truly caught up, and now everyone, including the FBI, knows that Hannibal is the Ripper, even though he must have had fake passports and has already escaped with Dr. Du Maurier.
Not yet aired. But all characters know Hannibal is the Ripper by now.
