What details in "Miss Brill" are important to focus on in order to help us understand Miss Brill's character and lifestyle?
I think one of the most important and most significant details we are provided with is when she is sat in the park and she begins to imagine that the events she sees are some kind of entertainment or drama that even she is involved in. This helps her to believe in a world where she has meaning and is significant, as opposed to being a lonely old woman who is ignored and looked down upon by the world. Note what Miss Brill believes:
They were all on the stage. They weren't only the audience, not only looking on; they were acting. Even she had a part and came every Sunday. No doubt somebody would have noticed if she hadn't been there; she was part of the performance, after all.
Miss Brill is a woman who craves significance and meaning, and wants desperately to believe that she is valued and important. The fiction she creates about the park being a play that she has a role in allows her to believe, however fleetingly, that she is important and that her absence would be noticed. However, when she returns to her dingy little room, "like a cupboard," by the end of the story we realise that this is a desperate attempt of a desperate woman to make herself believe she is significant.