I believe that there were two personalities in Azula. There was the normal teenager, who was somewhat domineering. This side was evidenced in the way she embraced her friends in Return to Omashu and her fretting over boys in The Beach. And then there was the dark and maliciously twisted side, which is evidenced all over the series.
When Azula lost Mai & Ty Lee, the normal side of her disappeared. This gave the twisted side room to completely take over. Now, excessively paranoid, Azula can trust no one, not even herself, as seen in her inability to manage and subsequent removal of her hair.
Seeds were planted in Azula's youth that led to this: 1. Ozai wasn't much of a father, he only used her to get ahead because she was a prodigy. Zuko complained about not receiving Ozai's love, but Azula didn't either. All she had was his favor. 2. Ursa's obvious doting on Zuko. This is typical of mothers who feel the need to protect the more sensitive child. Zuko was scorned by Ozai and so Ursa tried to compensate. But for the young Azula, it meant that mother loved Zuko more. 3. Then when Mai said "I love Zuko more than I fear you" the critical words were I love Zuko more. So yet another person chose Zuko over Azula. 4. The last person we expected to betray Azula was Ty Lee, but her friendship with Mai was apparently more important. The betrayal of Ty Lee told Azula that Ty Lee loved Mai more than Azula.
And we saw the results of this. In the mirror, Ursa told Azula "I love you, you're my daughter", but to Azula this must have seemed almost to mock her, especially since the whole scene was imaginary."
I was comparing our faces... haha, whatcha think?
Trying to be my very best Azula!