Disclaimer: I don't hate Snape, but I do dislike it when people misjudge and get his character wrong. He was a grey one (Rowling herself stated this), and not some white knight in shining armor.
No, despite what the film tried to show with sappy Sev getting all emotional, he never cared about Harry.
All Snape's ever done, including being spiteful towards Harry, was due to Lily having been killed. Snape also saw James whenever he saw Harry, and don't follow the film's version where he says 'you have your mother's eyes'.
That never happened in the book (primary canon).
Even though Harry was said to have inherited Lily's eyes, it was more his James resemblance that always irked Snape.
Right up until the end, Snape never ever warmed up to Harry, contrary to what Dumbledore had hoped for. Here's another difference in the misleading dialogue of the final film: Dumbledore was NOT as angry and miserable as his film counterpart. In the book, Dumbledore got happy and excited when Snape said the Pig for Slaughter thing. Dumbledore thought that Snape was finally warming up to Harry but, alas, Snape didn't give a damn about the kid.
Proof? Here we go:
‘I have spied for you, and lied for you, put myself in mortal danger for you. Everything was supposed to be to keep Lily Potter’s son safe. Now you tell me you have been raising him like a pig for slaughter –’ ‘But this is touching, Severus,’ said Dumbledore seriously. ‘Have you grown to care for the boy, after all?’ ‘For him?’ shouted Snape. ‘Expecto patronum!’ From the tip of his wand burst the silver doe: she landed on the office floor, bounded once across the office and soared out of the window. Dumbledore watched her fly away, and as her silvery glow faded he turned back to Snape, and his eyes were full of tears. ‘After all this time?’ ‘Always,’ said Snape.
There we go; it was always about Lily. He even calls Harry 'Lily Potter's son and refuses to try and warm up to the child.’
Snape never 'cared' about Harry at all.
You see, in terms of caring for Harry, that was Dumbledore's thing throughout Harry's time at school.
For even a man who grew up on 19th-century principles with a dysfunctional family, and who lost his sister, still cared for Harry until the end.
Some of you might argue that Dumbledore was manipulative and all that—and I don't completely disagree. But, if we are to be talking about caring for Harry (as Lily would have wanted), at least Dumbledore acknowledged the poor childhood of Harry (he does so in book 6) as well as being kind to the boy in school.
In the books, most of Harry's aggressiveness towards Snape (which included going into the Pensieve and invading Snape's privacy) was due to Snape's hostility from day one.
Dumbledore could at least be justified for having left Harry at the Dursleys, because anywhere else (except with Dumbledore himself—who’s not exactly fit to raise kids, really) would be too dangerous.
Imagine if Harry grew up being spoilt rotten in a wizarding family who enjoyed the benefits of having the famous kid under theiroof... Damn, that might lead to Harry becoming one helluva spoilt, manipulative brat in school. Also, there would be no blood protection in any house except that of Petunia Dursley (née Evans).
But Snape had no reason to be such a petty jerk to an eleven year-old child from day one at school. So what if Harry got nervous and couldn't answer those three questions? So what if Neville melted his cauldron? Snape went and attacked Harry for all of that.
'Pig for Slaughter', my arse. Snape didn't know the real truth behind what Dumbledore knew—that Harry would survive due to Lily's Protection having been taken in (unknowingly) by Voldemort when the latter used Harry's blood in the graveyard.
You see, while Voldemort had bypassed some of the blood protection by using Harry's blood, he also made the mistake of sealing his own fate.
After Harry's fourth year it was then the following:
So long as Voldemort lived, Harry could not be killed by Voldemort himself.
And this is why book!Dumbledore said that Voldemort was so knowledgeable and yet so hopelessly ignorant with regards to magic. This is how the books showed Dumbledore's superiority (both in his knowledge and the Ministry duel… where he deftly kept Voldemort at bay until Fudge arrived. Dumbledore wasn't struggling like his weak film version… no disrespect meant to Michael Gambon, though).
In a strange sense, we can almost say that Voldemort having taken Harry's blood made him like a ‘Horcrux’, of sorts, to Harry against Voldemort himself.
Here's book!Dumbledore explaining it at the limbo between the living and afterlife worlds to Harry:
‘He took my blood,’ said Harry. ‘Precisely!’ said Dumbledore. ‘He took your blood and rebuilt his living body with it! Your blood in his veins, Harry, Lily’s protection inside both of you! He tethered you to life while he lives!’ ‘I live … while he lives? But I thought … I thought it was the other way round! I thought we both had to die? Or is it the same thing?’
That's why the Killing Curse in book 7 could only kill Voldemort's piece of soul within Harry. It had nothing to do with the Elder Wand in the forest scene.
Dumbledore guessed all this, and that's why he appeared triumphant after Harry told him of the graveyard ritual. There was now hope for Harry to survive Voldemort's attacks.
Anyway, back to poor Sev:
Snape never ever gave a damn about James' son; it was always about avenging his fallen childhood infatuation. The very same woman who Snape blurted the Mudblood word against (oh yes he did, which went to show his innermost mindset) in their fifth year.
Snape had a mission and got it done; that was all.
NB: Comments disabled because, despite my usage of book canon and plenty of facts, there will always be those diehard Snape whitewashers coming in for the attack. And because I praised Dumbledore in this answer, there would also be an influx of WAAWAA DUMBLEDORK WAS EVIL!!!
And let us not forget the whole package of:
SNAPE LOVED LILY! JAMES WAS EVIL! SIRIUS WAS EVIL! JAMES CAUSED SNAPE TO BECOME A DEATH EATER! SIRIUS CAUSED JAMES TO CAUSE SIRIUS TO CAUSE MOONY TO CAUSE SNAPE TO BECOME A DEATH EATER!!!!!