Poe was a master at creating suspense. He is able to create suspense by introducing us to an unstable narrator (what we can an unreliable narrator) who clearly is mentally ill. Because of this, the reader does not know what to expect from him. This definitely builds suspense when the reader can bet on the unexpected from a narrator.
One of the main events that builds suspense is when the narrator watches the boarding house owner at night, staying still for the entire night. The reader is there with him while he does this and while he plots to kill the boarding house owner.
Also, Poe uses stream of consciousness in this story and as the narrator lets the police into the boarding house, the narrator's thoughts become more and more fragmented until the narrator screams his admission of guilt.