To write an essay about Emily Grierson as an unsympathetic character, the first step would be to go back through the story and mark notes where the narrator makes her appear unappealing. For example, in the third paragraph, the narrator says that she was "a duty, and a care; a sort of hereditary obligation upon the town." What he doesn't say is that she was a pillar of the community or a beloved elder. There is no evidence to suggest that she did anything philanthropic to help the town or its citizens. Other negative traits that the narrator describes include her arrogance, seen in her dismissal of her tax bill and refusal to meet with the mayor.
Emily Grierson's apparent murder of Homer Barron is the act that most resoundingly makes her unsympathetic. In a paper that analyzes her character, it would need to be a prominently developed section.