Snape has always treated Harry with pure and utter loathing during the whole Harry Potter series and always looked for a chance to put him down during class. But was his helping/saving of Harry for Harry's sake, or for the sake of helping Lily's son?
Yes, he did
The whole purpose of Snape's actions throughout the series is to protect Potter. From the first movie (where he tries to neutralise Quirrel's spell during the Quidditch game) to the agonised conversation he has with Dumbledore about Potter's fate confirm this.
It could be that he is protecting Potter because that is the only way Voldemort can be defeated but his motivation for that is because Voldemort killed Potter's mother. We could suppose that his motivation was just to get revenge for the murder of Lilly and caring for Harry was just a means to that end. But his last living act–he donates his memories to Harry so Harry can understand his motivation–suggests he wants harry to understand why he behaved the way he did towards him. Those memories also explain some of the motivation for his harshness with Harry: Harry has some of the less noble characteristics of his father (who stole the Love of Snape's life from him and who was sometimes a bully to Snape). So he loves Harry because he is Lilly's son but hates some of his characteristics because he is also his father's son. So he cares for him but also has some reservations about his character.
Moreover, for his role with Voldemort to be convincing, he has to act as though he dislikes Potter. He can pull this off convincingly partially because of his dislike for Harry's father and Harry's behaviour when he emulates his father.
Those mixed motivations show that, ultimately he cares for Harry, but not without reservations.
In a way, yes. Mostly, especially to start with, he only took care of Harry for his mother, Lily, the love of his life. Towards the end though, especially that last dying scene, I believe Snape did begin to show affection for Harry.