The answer to this question can be found pretty quickly in the beginning of the story itself, if you read closely. The narrator, widely assumed to be a man, states in the second paragraph rather plainly that he had no real motive for killing the old man. He tells us that
"Object there was none. Passion there was none. I loved the old man. He had never wronged me. For his gold I had no desire."
So, the old man had never made him upset, so he didn't want to kill him out of revenge or anger. The narrator also says he didn't want to steal the old guy's money, and there was no "object" or purpose for his motives. No passion, no intensity of hatred against the poor guy. Then, he gives what is possibly the strangest reason in any murder tale, for wanting to kill the old man. The narrator says it was because
"He had the eye of a vulture—a pale blue eye, with a film over it. Whenever it fell upon me, my blood ran cold; and so by degrees—very gradually—I made up my mind to take the life of the old man, and thus rid myself of the eye forever."
Bizarre reason to kill someone, if you ask me. The old guy's eye bothered him. Maybe the man was blind, or had an eye condition, but it looked a bit creepy--milky, with a film over it, and that eye bothered the narrator so much that whenever he saw it it made his blood run cold. So, his obvious conclusion is that he must kill the old guy, just to get rid of the eye. It's an interesting reason, one that begs the question of the narrator's sanity. I hope that helps a bit; good luck.