What are some particular techniques that the author uses to develop Miss Brill's character?
Mansfield uses a simile to begin the story, in order to show us Miss Brill's initial state of mind: "the blue sky [was] powdered with gold and great spots of light like white wine splashed over the Jardins Publiques [...]." This simile, comparing the light in the sky to dashes of white wine, conveys Miss Brill's impression of her own worldliness and vibrancy.
Further, Miss Brill personifies her old, dowdy fox fur, imagining that she's "rubb[ing] the life back into the dim little eyes" and that the fox speaks to her, asking, "'What has been happening to me?'" with its "sad" eyes. Readers might begin to wonder at this point if the fox is somehow symbolic of Miss Brill herself. Just like the fox has been tucked away, growing old in its little box, so has its owner. In the end, Miss Brill returns to her "little dark room -- her room like a cupboard," like the same rooms that she's imagined the other old people at the park to have come from. Likewise, she imagines the fox to be asking the question that she seems likely to have asked herself, at least on a subconscious level: what has been happening to her? Not much. She's been growing old, growing obsolete and out of date, just like her fur. Miss Brill convinces herself that the fox is a "Little rogue" because she so desperately wants to attribute such vitality, still, to her own self.
One of the most prevalent techniques that Mansfield uses is stream-of-consciousness here. She also uses description of the character's actions to reflect the character, as well. Mansfield does a masterful job at relaying Miss Brill's thoughts to her readers in this stream-of-consciousness narrative. Miss Brill's mind drifts to other thoughts as she observes others in the park, much as our own minds might drift to memories, etc., if we were in her position. Also, the way that Miss Brill reacts to the young couple making fun of her and her fur is indicative of her character, as well.
Irony is used extensively. Nearly everything Miss. Brill states about the characters around her such as they old, odd, and look like they come from their cupboards is indicative of how others see her. Unfortunately, she does not realize it.
In addition, the author uses music and imagination. Miss. Brill sees the band as the sound track of her life so to speak. From that notion, she begins to believe she is an actress in a play. This "revelation" gives Miss. Brill a false sense of connection with those around her and false view of herself. It also reveals how odd, lonely, and out of touch she is with herself and her surroundings.