The eponymous Beast from Disney's Beauty and the Beast. He starts out seeming like the monster he appears to be, but we soon learn the anguish he feels about this behavior and how it's prevented him from becoming normal again (he even has a song about it in the Broadway version). His interaction with Belle then steadily pushes him into the realm of a Jerk with a Heart of Gold.
The DVD commentary discusses it in great detail, citing that no matter how bad he is your sympathy must always be with the Beast in order for the movie to work, and thus a lot of emphasis is given the constant emotional pain he lives in.
The eponymous protagonist of Coraline starts off the movie being essentially a self-centered brat, although, arguably, she does have a Freudian Excuse in her distracted parents. She also goes through a lot of heart-wrenching crap: Having to deal with the Other Mother wanting to eat her and keep her for herself and wanting to sew buttons on her eyes as well as nearly losing her parents and losing the Other Father and Other Wybie.
Nuka, however qualifies. His father chooses an unrelated cub over him, he's portrayed as The Unfavourite to his mother, and seems to be the Butt Monkey of Zira's pride. While he does want power, he also wants to gain the favor and attention of Zira, who is more concerned with Kovu. In the end, he dies trying to prove himself to his mother, and even fails at that. He spends his last breath apologizing to Zira for failing, his death prompting one of the only displays of love and affection that Zira probably ever showed him.
Mirage from The Incredibles. Sure, she had a hand in leading numerous superheroes to their deaths, but she's against killing children, nearly dies twice, and her supposed boyfriend, Syndrome, is shown to not care about her when he sadistically goads Mr. Incredible into trying to crush her to death.
Up: Charles Muntz - rejected by the scientific community, eluded by a stupid bird for sixty years, all by himself, going completely insane...it's so tragic...Of course, murdering innocent people because he thought they would take his bird away makes him a bit of a monster.
Shrek: Prince Charming. After enduring grief and humiliation now that he no longer has the power his mama gave him, all he wanted was his own Happily Ever After... which he probably would have gotten had he not stayed a villain by choice.
Perfect Blue: Rumi. She ends up delusional and tries to kill Mima. Me-Mania too. Yes, he was a serial killer and attempted rapist... But that's because he was a deformed, barely-functioning schizophrenic who lived only for his perfect image of Mima. The scene where we see his apartment is an enormous Stalker Shrine and that all he wants in life is "the real Mima's" affection was genuinely sad.
The eponymous character of Megamind. On one hand, he's an Ineffectual Sympathetic Villain who regular kidnaps someone, threatens them with bodily harm and always tries to kill his nemesis, Metro Man. On the other hand, his planet was taken away by a black hole, along with his parents who can't escape from it, he grew up in prison, and was ostracized in school by the kids.
Princess Atta from A Bug's Life. A soon to be Queen ant who's acting in a way that is out of a severe inferior complex. She's rather strict towards her own younger sister, Dot. Not to mention refusing to see Flik's inventions as a good thing to help the ant colony. But she is rather nervous about her problems with taking over as a Queen that you're amazed she doesn't go further.
The ant colony itself as a whole. Sure, they brought the Darkest Hour on themselves by ostracizing Flik and PT Flea's entire circus after they hear the truth about the "warriors". But do they deserve the treatment Hopper gave to them afterwards?
Dave from Penguins of Madagascar. Once living at the Central Park Zoo, he had a roomy habitat and all the fans. Then started a chain where penguins unintentionally stole his glory as the octopus was moved to zoos and aquariums around the world, where he was forced to live in small tanks.
Prince Hans from Frozen. See his backstory for details and motivations of his actions. (i.e. growing up in a loveless family)