I already put the pieces together but I still wish we could have gotten more detail about it

Definitely. She deserved more back story than she got.

all she got was The Beach which is probably my least favorite episode. it is so out of character for the show. instead of taking the opportunity to explain Azula's angry personally they just reaffirmed her craziness. they almost had it with "my own mother thought I was a monster" and then they lost in with "she was right of course". :P

You guys should read the epilogues. They're all over youtube. Each is in three parts, and their names are The Promise, The Search, and then there's another one that's not out yet and I can't remember the name... anyway, they go deeper into Azula's character more there.