Here's what we know for sure about Homer Barron:
#1. Miss Emily kills him with rat poison. Oof.
#2. The Jeffersonians don't like him much: he's a rough-talking, charismatic Northerner.
And...that's about it. Everything else we can say about Homer Barron is conjecture. But, like the people of Jefferson, we love to speculate.
We don't know how involved Homer was with Emily—he may have intended to marry her, but became dissuaded by the wacky antics of her cousins and the town. We don't know why he went to her house that last time, or how exactly his death took place. We also don't know if he liked women or men.
The following line suggests that the people of Jefferson had suspicions about his sexual orientation:
Then we said, "She will persuade him yet," because Homer himself had remarked – he liked men, and it was known that he drank with the younger men in the Elks' Club – that he was not a marrying man. (4.1)
Here, the town seems to be saying that, even though he isn't the marrying kind, Emily might still manage to hook him. The gossipy tone of the narration seems to imply a courtship of convenience: Emily is over thirty and therefore undesirable, and Homer is (potentially) gay. Both Emily and Homer are being negged pretty hard here: the townsfolk are suggesting that only a gay man looking for a beard would consider Emily, and that only a desperate spinster would consider a man who "liked men."
Of course, this could all be false.
After all, Homer's a Northerner, and therefore worthy of scorn in the eyes of the Southern Jeffersonians. This ingrained prejudice could have also contributed to the fact that Homer Barron's disappearance didn't provoke much of a fuss in the community of Jefferson. Even though Miss Emily's house started to smell like rotting flesh after Homer Barron seemingly skipped town, the community didn't try very hard to figure out if there had been any foul play.