The narrator lives in the same house as the old man, at the beginning he tried to prove he is not insane to the reader regardless he was going to kill the oldman.
I killed him!
The reason of killing the old man was simple. The narrator said it was the old man's vulture eye.
The narrator entered the old man's bedroom over seven night. Waiting for the chances to get rid of the eye, so it wouldn't trouble him no more.
On the eight night, the old man woke up, but the old man's time had come! With a shriek, the old man was pulled on the floor and killed by the narrator.
After the narrator hide the corpse violently. The police come and said the neighbor heard a shriek come from old man's house.The narrator welcome them and show them around, He even let them sit and chat for a while.
Then the narrator heard low heartbeat, It grew louder and louder, The narrator thought it was the old man's heart The narrator showed his insanity and got arrest by the police.