How does Iron Man's costume serve to emphasize the aspects of his character?
Now first of all, we have to take a look at the functionality of the costume (or rather suit), which obviously is the creation of a genius scientist and inventor. In fact Iron Man is pretty much the Marvel equivalent of Batman (though, less brooding and more self-centered, and with a war trauma as incentive instead of the loss of his parents), whose strength comes from his intelligence combined with the wealth necessary to put his ideas into actual form. He prevails through a sharp mind and technology, rather than superhuman strength or otherwise alien abilities. And according to this related answer this technological superiority of his suit also supports the techno-fetishism from which those movies and their protagonist derive much of their fascination.
So much to the mere functionality. But afterall there is a much more subtle or trivial aspect to this costume, namely the fact that it is exactly that, a costume/suit and not an autonomous robot or a remote-controlled drone. As the later movies, especially the most recent Age Of Ultron, show, Tony doesn't have much of a problem to just build androids and AIs doing his job. But as much as Iron Man's success might depend on a human controlling his behaviour, there is a much more indirect advantage of a suit, it puts Tony Stark in the middle of all, which this fits to Tony's incredibly self-centered and arrogant rock star character. It is Tony Stark who does all this, he is the Iron Man, as he is more than ready and proud to admit and exhibit throughout the movies, starting with his "outing" at the end of the first Iron Man movie, and repeatedly emphasized during his show acts at Stark Expo, his reluctance to share his invention (though, that might also have other motivations), or his "cool" introduction in the first Avengers movie.
And this rock star aspect is also emphasized by the suit's most prominent visual features, its colors. As shown in the first Iron Man movie, this red-golden coloring is utterly useless and the only reason he adds them is because they look good and this yet again supports his extroverted and expressive character, seeing that red and gold support notions of power, excellence and an overall hot temper. After all, as this related question and its answers show, the colors of the suit are directly related to Tony's love for Hot Rods which are in turn yet another facet of his love for machines and his excessive "live fast, die young" attitude.
Tony Stark is one of the most well developed characters not just in the Marvel Cinematic Universe, but in modern cinema today, and his relationship with his Iron Man suits is as fascinating as it is unhealthy.
If you recall the events of Iron Man, you will remember that Tony Stark first designs his Iron Man suit (and later refines it) in order to escape captivity in Afghanistan, where he was being held by the Ten Rings terrorists. Tony was blown up, tortured, and shown his life work was in the hands of his enemies, before being forced to kill multiple people in order to escape - an escape that cost Yisen, the man who saved him, his life. It doesn't take many rewatches of Iron Man and Iron Man 2 to see that not long after his imprisonment, he begins to suffer from PTSD (post-traumatic stress disorder) - simply look at his actions in the films, which see him shut down his companies main division, completely change his personality and begin illegally entering active war zones due to his new found need to help people, almost all of which are motivated by a greater need to "take control". We also see Tony putting himself in harms way and acting recklessly, which is also a symptom of PTSD.
In relation to this PTSD, it should be fairly clear that Tony's armor isn't just a weapon or a superhero costume - it's a safety blanket, a way for Tony to feel safe. Although subtle at first, we see Tony's symptoms of PTSD getting worse in Iron Man 2, and the revelation that his arc reactor is poisoning him (causing him to face his own mortality yet again) results in a near total breakdown, which sees him choosing the stay in the Iron Man suit even when it isn't really needed.
By the time Iron Man 3 rolls around, Tony Stark is in a worse state than ever after flying through the portal in The Avengers. A panic attack caused by a child asking him about the Battle of New York sees him running through a crowded restaurant just to get into the nearest Iron Man suit, his increasing dependence on the suits putting his relationship with Pepper Potts under strain. Iron Man 3 ends up doing a lot of work to help Tony 'heal', or so to speak, forcing him away from his suits in order to show him that he doesn't need them as much as he thinks - and by the end of the film he realises that it is he who is Iron Man, not the suits.
But that doesn't change his general outlook on the world and the dangers it faces. In Avengers: Age of Ultron we see Tony attempt to create an AI that can replace the Avengers - and in his own words describes it as;
A suit of armor around the world.
In short, Tony's Iron Man suits serve to emphasis the damaged state of the character, his need to feel safe and his growing sense of unease as the world around him grows more dangerous.