It would be easy enough to blame Emily Grierson's father for all of her troubles, but Emily did little to better herself during her life. Yes, Emily grew up under the thumb of a domineering father who repressed her and discouraged her male suitors, any one of whom could have taken her away from the father and the troubles that she would inherit from him. After her father died, she was left with a crumbling old house, little money, and no prospects for the future. When Homer Barron arrived, she leaped at what she must have known might be her last chance to begin a new life. When he decided to leave her, she must haved decided that life with a dead body in the house was better than none at all (remember, she had already come to the same conclusion with her father's body after he died).
I have only limited sympathy for Emily, since she was an adult who should have understood the implications of her actions. She was left scarred by the deeds of her father, and for this she should be pitied; but the woman she became following his death was guilty of anti-social behavior, elitist snobbery and cold-blooded murder--hardly characteristics deserving of overwhelming compassion.
To me, Miss Emily Grierson is someone who deserves our sympathy. Of course, killing Homer Barron was not the right thing to do, but I would argue that we should sympathize with her because she was driven to the killing by her father -- it is not something she probably would have done if she had been raised in a normal way.
Her father kept chasing away men who wanted to marry her. He thought that they were not good enough for her. Because of this, she is very lonely and desperate. Because she is lonely and desperate, her solution to Homer trying to leave her is to kill him.
It's surely her fault that she does this, but ultimately, I think it is her father who was put her in a terrible situation. Therefore we should pity her.