Azula is one of the most complicated characters in the Avatarverse. She is presented as a ruthless powerhungery girl who will use anyone to achieve her goals. We aren't presented with a whole lot of background information on her.
Most of what we get is shown through Zuko's flashbacks. She and Zuko were raised very differently. From what is shown, Ursa cared for Zuko very much and would do anything for him. Her interactions with Azula aren't many and normally when they are together Ursa is scolding her. Azula felt that she didn't get enough attention from her mother. She possibly felt that getting negative attention over positive was better than nothing.
If we run with the idea that Ursa did favor Zuko, then Azula was left with her father to give her guidance. Ozai does not seem like the type of man to give a child the affection they need. He clearly looked down on Zuko, declaring him weak, a view that could have easily been passed onto an impressionable child. He is a man that would most likely desire perfection, something Azula strives for, declaring that "Almost isn't good enough". (201) He wanted the throne for himself and these ideas passed onto Azula.
Ozai is a man who had no problem scarring and banishing his son and earlier on planning to kill him, should have any reason not to doubt that he could have been cruel to both of them? They are different people though, Zuko and Azula, and would have had different reactions to their father's treatment. Where Zuko could confide in his mother, Azula turned inwards. It has been shown that children who are abused or mistreated by their parents often become abusers. And what is Azula if not an abuser?
Azula has serious issues with her mother. She feels that her mother loved Zuko and not her, that she thought she was a monster. Azula admits that this hurt her and that shows that she is not the hardened woman she makes herself out to be. During Sozin's Comet she has visions of her mother. The hallucination tells Azula that she loves, her, that she, Azula is confused, and that she doesn't think that she is a monster. These are things that Azula, deep inside, wishes her mother would say to her. She doesn't believe it is possible though and the hallucination mocks her and in anger she shatters the mirror.
When she and Zuko receive gifts from Iroh, Azula is clearly angry and jealous that Zuko got a knife with some sort of meaning to it while she got a doll, something that in my opinion, she's have no interest in. She would possibly even take this as an insult, that she should act more like a young lady.
Azula has only two friends: Mai and Ty Lee. Ty Lee is a very affectionate and emotional person, something Azula isn't used to. During the party in The Beach, Azula tells Ty Lee that the boys don't really like her. Once Azula sees that she's upset her friend she awkwardly tries to make Ty Lee feel better. Ty Lee seems to be the only person who's feelings matter to Azula. During The Beach episode, Azula is shown as having no social skills. She can't function in a normal situation. She keeps returning to what she knows, war, in order to cope. A game of 'volleyball' literally becomes a battle to her. She is later shown not knowing how to act around boys and does not like the idea of acting like an idiot to make them like her. When Mai, Ty Lee, Zuko, and Azula sit around the fire sharing their feelings, Azula at times reacts hostily. Showing your emotions are a weakness in her mind.
Mai and Ty Lee's betrayal is the beginning of her downfall. I've heard of their friendship being referred to as a pyramid. Azula is at the top. She is their leader who strives to be the best. Mai and Ty Lee are on the bottom. When Mai and Ty Lee leave her she was left with no support. She may have acted cold toward them, but they clearly mattered to her. Mai turning on her was not nearly as shocking to her as Ty Lee. Mai left her for Zuko and then Ty Lee betrayed her as well. Zuko was given the love and affection. When Mai declares that she loves Zuko more than she fears Azula, it is a critical blow. Once again Zuko has taken something that she craves, something she may not even be aware that she craves.
While Azula may not have been a normal child to begin with, the way she was raised did her no favors. There are plenty of problem children who are able to live normal lives thanks to the help of their parents. Ursa doted on Zuko while Ozai bred Azula to be a weapon. A reunion with Ursa would be a huge step in Azula's recovery. She would probably never reach the level of being a hero or a selfless person, but she could possess the capacity to be good.