I feel like this is a great question, actually. As I trek through my abnormal psych class and we touch on this topic, I will keep Azula in mind.
See, one of the things I love about the Tyzula ship is that in terms of friends, Ty Lee is like the only one she actually gives a damn about. Like, yeah she had Mei too, but with her, she didn't really care as much what happened between them like she did with Ty Lee. At least, not that I saw. People like to say Ty Lee's betrayal is what finally made her snap; she lost like the last real love in her life. I mean, her father's love was obviously not real, but she had a hard time grasping this. We still don't know if she ever figured that out.
But, in general, in regard to others, I really do think she developed abnormally in terms of the mind. Yes, her father played a great hand in this because you can argue that he too is also mentally abnormal (of course this played on how he was raised, too. He had no way to know what a different lifestyle would be like).
I can't wait to see what they decide to do with her in The Search. Perhaps since it may not be a chemical imbalance that makes her mentally abnormal, they can correct the damage her father has done. Because, if her mother had been able to grab her and take her under her wing, perhaps she would've been more like Zuko. But, since her father got to her first and had such a strong grip, we have this Azula. So, hopefully it's reversible.
Great question! :D