William Faulkner makes a deliberate choice to begin this story at the end in order to build and maintain suspense. It is up to the reader to unpick what is going on and establish what the true order of events might be. Essentially, however, it goes something like this:
Emily Grierson lives with her father, who thinks no man is good enough for her.
Emily's father dies. She at first refuses to let anyone dispose of the body. For a while, after this period, she ventures out very little, and employs a "Negro man" to keep her house for her. A dispensation is granted for Emily, on the death of her father, saying she need no longer pay taxes.
Emily is sick for a while. Then, about two years after Emily's father dies, she cuts her hair short and reappears. A construction company arrives with Homer Barron as foreman. Emily begins to be seen with Homer Barron, and it is thought that she might marry him.
Emily goes to the druggist to buy some poison. Emily says it is for rats and people think she is going to kill herself with it. However, she doesn't—instead, she begins to order items for Homer, including clothes and a toilet set.
Then Homer Barron disappears.
A strange smell is observed coming from Emily's house. Rather than challenging Emily about it, some men cross Emily's lawn and put down lime to get rid of the smell.
Emily hides herself away and is little seen for six or seven years, when she begins to give lessons in china painting.
The city authorities come to Miss Emily to tell her she needs to pay taxes. Emily sends them away.
Emily dies, and the old black servant lets the people into the house. The body of Homer Barron is then found in a bed in her home.
We can infer, then, that Emily kills Homer before the smell is detected.