Faulkner chooses to tell the story out of order for reasons of building and maintaining suspense, so the first death presented to us is that of Miss Emily Grierson herself. When Emily dies, everybody in town goes to the funeral, partially to gawp at this town institution.
Next, it is mentioned that Emily's father died thirty years before, after which Emily did not venture out of the house very much. She had refused to relinquish the body at first.
The most significant death in the story, arguably, is the death of Homer Barron. This is alluded to from various angles; we can piece together that Emily probably bought poison and killed him, then kept him in her house, with this causing the smell around her house. At the end of the story, his body is found in Emily's bed.
There's another death which is important to Emily. Colonel Sartoris had decreed that the family should pay no taxes in the town; after he dies, Emily is forced to argue her case on this point.
The only other death referred to obliquely is that of Emily's great aunt, old lady Wyatt. She went "crazy" at the end, and Emily's father had fallen out with the rest of the family over this woman's estate, leaving Emily without an extended family willing to speak to her.