Characterization is a detailed description of a character’s physical and personality traits. You have chosen an excellent story to ask about characterization, because this story is a perfect example of indirect characterization, which is the process of describing a character indirectly through the character’s thoughts and actions, rather than telling us directly about the character.
We know several things about Miss Brill from her visit to the park. First of all, we know that she is a people watcher. She enjoys watching people as if they were a show. We also know that she cares about people. She is a teacher, which is a service profession, and she reads newspapers to old people. Both seem to indicate she is selfless and giving.
Miss Brill is also a bit naïve. She is unmarried, thus the miss, and she doesn’t seem to have any romantic relationships. She gets so caught up in watching the people, especially the young couple on the bench next to her, that she forgets that this is real life and she is eavesdropping. She means no harm though. She lives largely in her head.
Miss Brill is sensitive, but she does not really recognize the emotion. She loves her park, she loves watching people, and she loves her necklet almost to the point of imaging it has feelings. When she is ridiculed, she does not really internalize the emotion and recognize that she is, indeed, sad.
Ms. Brill is a expatriate British teacher living in France, presumably a spinter, who lives in isolation with the exception of sporadic visits to an elderly friend to whom she reads. Her visits to the park on Sunday, and her connection to the fur necklette she wears make you realize that she is alone in the world, with not much to do, since she has spent her life in isolation, consoling herself with the simplicity of park visits, and observing people.
Due to her lack of social interaction, she has internalized a world of her own, where she gives roles to each of the strangers she sees in the street, perhaps as an attempt to connect with them in some internal way as she is unable to do so physically.
She is obviously not happy with her life, though she makes herself believe that she is. This is documented in the way she metaphorically shows how her fake necklette, which she is so proud of, is "crying" as she puts it back it its isolated box. She was crying too, internally, as she put it away since she felt (through the mockery of other park visitors) that she may not be at all welcomed in the world. Maybe, she feels, she must remain alone. It is her reality.
Ms. Brill is the epitome of the human being hungry for connections and communication. She is incapable of changing her situation, so she succumbed to it. Her nature is undeniably a call for help, but nobody seems to listen.
Miss Brill is an English teacher, presumably to middle or secondary school students. By all accounts, she is good at her profession. Teaching occupies a great deal of her life, not because of a particular passion or zeal but more because there is little else in it. The weekends, in particular Sunday, allow her to indulge in her once a week routine of taking out a prized and revered fur necklet and sitting in the park and, essentially, engage in people watching. Her engagement in this activity allows her to believe that she is an integral and essential part of this community setting. She believes that band plays music that she enjoys, that the people in the park are empty without her presence each and every Sunday, and with her fur, represents the center of this small communitarian universe. Within this setting is a strong sense of denial of reality, for she is committed to the power of her self imposed illusion. Her connection to this world is an illusion, and Miss Brill's inability to distinguish between what is real and what is in her mind represents a great deal of this character. In Miss Brill, the reader sees how dreams and illusions can be both motivating, but also painful, as one is crushed under their weight. Certainly, Miss Brill, as she blames the fur for the climax of the story, seems to be pinned under such a burden.
For starters, give a physical description of the character. She's middle aged and thinks that she is well dressed and fashionable. This initial presentation of her is merely a facade and we learn that she is a rather tired and deluded character.
Another area to look at is her clothing and what this represents about her. Her personality is directly reflected by what she wears and her internal dialogue shows us that Miss Brill verges on arrogant.
By the end of the story, she realises that she is far from a glamorous player in the game of life, rather she is insulted by a young couple who look down upon her. She is dull and alone.
It is clear that Miss Brill leads a very restricted life. We are not told that she has any friends, and her main connections with life are two. The first is an "old invalid gentleman" to whom she reads the paper, and the second is her eavesdropping on the conversations of fellow citizens who come to the park on Sunday afternoons. She comes regularly to the park, and indulges herself in her characteristic fantasy that she is a performer, along with all the others there, in a vast human drama. She thinks of some of the older people as having no life at all (paragraph 5, page 178), and it is not until the story’s end that we realize she is actually describing herself, for she is just like these other people.