In To Kill a Mockingbird, why was Alexandra really staying with Atticus, Jem and Scout?(chapter 13)
Aunt Alexandra has really come to Maycomb to stay with the Finch family because she does not think that Atticus is raising his kids properly. She thinks that they are not acting in a way that is appropriate for members of a high class family. She also thinks that Atticus is giving Calpurnia (who is, of course, black) too much of a role in raising Scout as she gets older.
You can kind of see this idea right from the beginning of the chapter. Just as soon as the chapter begins, Aunt Alexandra is telling Scout to quit scratching her head. Later on, we start to hear about how Aunt Alexandra believes in some families just having better bloodlines than others.
So Aunt Alexandra has come to Maycomb to try to reform the Finch family and get them to act more in accordance with their proper place in society.
From what i've gathered from the text, the finch blood line was relativly wealthy before the 1930's depression hit and that wealth installed an acute sense of pride in Aunt Alexandra. She enjoyed mentally placing herself high on the social scale because of it. This pride was substantial enough that even after the depression took her family's money, she found pride in the fact that her family once had it and this was what started her obsession with hereditry. It was how she was able retain her high social status in relation to others such as the Cunninhams who had always been poor. Atticus who is not a very proud and pretentious man (seen in how he did not tell his children about his skill with guns) would natrually not forced this sense of bloodline pride on his children. Aunt Alexandra was there to enforce the cultivated image of what it was to be a finch onto the children. Situation dictated that there was every reason for scout and Jem to behave like commoners but Aunt Alexandra was there to prevent that.