One word to describe the narrator is "nervous." The narrator admits this in the first sentence. He even claims he is "diseased" but adds that his disease is not madness; rather, it is something that has made his senses more acute. Clearly, he is mad and clearly, he is trying to convince himself and/or the reader that his hallucinations and extreme sensitivity to sights and sounds are actually benefits rather than curses of his disease. The first paragraph says it all. He claims that his sense of hearing was incredible. He claims it is not madness that made him hear/hallucinate the things he heard; it was simply an overly acute ability to hear. He is deluding himself. "Delusional" is another good word to describe the narrator:
Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad?
The narrator continues to try to justify that he is not mad by showing how calmly he carries out the old man's murder. The narrator claims to love the old man but then kills him because he doesn't like the look of his eye. In other words, the narrator says one thing and then does something contrary to it. He is intentionally or unintentionally (on account of his madness) duplicitous.
Some critics suggest that the narrator's obsession with the sound of the heart beating is not just madness and not just an hallucination derived from his guilt. It is also an obsession with time. The sound of the beating heart is like a ticking clock. The sound is therefore like the passage of time. The narrator is "obsessed" (another good word) with hearing, being aware of, the passage of time. Therefore, he could also be obsessed with death. As an additional interpretation, the narrator, after killing the old man, could be hearing his own heart. So, he'd been obsessed with the thought of his own death; in his madness, he thought killing the old man would kill this obsessive awareness of death in general and maybe his own death in particular.