By all indications, the story is told from the viewpoint of a third-person narrator; much of the story also highlights the perspectives of the townsfolk regarding Miss Emily.
The reason the story is not told by Miss Emily herself is because, at the time of the narration, Miss Emily has passed away. The story is told as a reconstruction of Miss Emily's life. From the viewpoint of the narrator, Miss Emily had lived a mysterious and tortured life; in fact, the discovery of Homer Barron's body in a sealed, upstairs room in Miss Emily's house had further validated this theory.
The story is told in five parts. In the first part, the narrator recalls how the townsfolk had attended Miss Emily's funeral. So, even from the beginning of the story, we can see that Emily is no longer alive. This first section of the story also describes how Miss Emily cements her powerful position in the town. Citing the deceased Colonel Sartoris as the authority for her actions, she dismisses the town leaders' demands that she pay her taxes.
In the second part of the story, the narrator describes a time when a strange smell had permeated the vicinity of Miss Emily's house. No one ever discovered where the smell had come from (at least, not until after Miss Emily's death). Later, it was said that Miss Emily, like her great aunt (old lady Wyatt) "had gone completely crazy at last." In fact, Miss Emily supposedly refused to bury her deceased father for three days. It was only at the behest of the ministers and the doctors that she relented.
The third part of the story recalls the time when Miss Emily was courted by Homer Barron, a day laborer from the North. The narrator also remembers the time Miss Emily had purchased arsenic to (supposedly) kill some rats.
In the fourth part of the story, the narrator details the disappearance of Homer Barron and Miss Emily's eventual death. The last section of the story describes Miss Emily's funeral and the shock experienced by the townspeople when they discovered Homer Barron's body in a sealed upstairs room in Miss Emily's house.
So, the story is told from the viewpoint of a third person narrator who, after Miss Emily's death, tries to provide an unbiased account of her story.