The story's exposition is delivered within its first two paragraphs. Here, the narrator provides the background information that we need in order to understand the story -- and his character -- and this exposition ends when the narrator says that he decided to kill the old man "and thus rid [him]self of the eye for ever." His decision to kill the old man because of the old man's "vulture eye" is the story's inciting incident, and it initiates the rising action.
During the story's rising action, the narrator explains how he stalked the old man, creeping into the old man's bedroom every night at midnight. Eventually, on the eighth night, the old man is awoken by a sound, and when the narrator sees the old man's "vulture" eye he grows "furious" and kills the old man. He goes on to dismember and hide the body beneath the floorboards and he even allows three police officers to enter, as they've come to investigate the old man's shriek, which was heard by neighbors.
The rising action continues until the narrator begins to hear a sound that he believes to be the old man's now-dead heart beating beneath the floor. This is the climax, the moment of the greatest tension in the story where its main conflict is most clear. He says, "No doubt I now grew very pale; -- but I talked more fluently, and with a heightened voice." He feels that the police can hear the sound he hears and that "they knew!" that he had killed the old man. To him, it feels like utter "agony," and so he confesses. The story then ends immediately after this climax and his confession, and there is no falling action or resolution.