I feel, that seeing as she was the eldest of her siblings, she was given less attention and had to do things on her own(<---speaking from personal experience here). Bellatrix may have been ignored by her mother and wasn't able to project motherly love to others because she had not experienced it herself, which would explain why she could kill without thinking twice about it. Also, Bella might not have felt as important as a child and grew up hating her childhood. I think she wanted to prove herself and make her parents proud of her.

Or, in another thought of mine, Bella was the star of her parent's sky and she was taken in to their muggle-hating lives and she wanted to follow in her parents footprints and make an impact, or so she thought, being raised the way she was. I think that she thought that being disrespectful to anyone who was not up to her or her parent's standards was how everyone was supposed to act. She was taught that wrong was right and right was wrong and Azkaban was no help in setting her straight.

Oh! I have another theory. Perhaps as a child she was given so much attention and all that jazz that she became so cruel and wicked. I think that her parents could have seen Andromeda being the youngest and most needy(of help doing things) as Andromeda being the weakest link and that is why she truned out the way she did.