The narrator's tone is defensive (which means, he seems to be defending himself in an argument). It almost feels like he has been accused of being a nervous and insane person, and his tale is his defensive response to that accusation. From the very get-go, he agrees that he is indeed a nervous person,
"but why will you say that I am mad?...How, then, am I mad? Hearken! and observe how healthily—how calmly I can tell you the whole story."
He insists he isn't mad, and in his defense, seems to be saying, "Looky here. I'm going to tell you this story, and tell it so calmly that after I am done, you'll know I am not mad...a crazy person wouldn't be able to tell you such a story so calmly." So his tone from the beginning is defensive and almost pleading for the reader to find him sane. He continues along this vein, inserting throughout his entire tale assurances such as:
"Now this is the point. You fancy me mad. Madmen know nothing. But you should have seen me. You should have seen how wisely I proceeded—with what caution—with what foresight—with what dissimulation I went to work!"
almost as if it is impossible for an insane person to work in such a wise, cautious and logical way. The reader is left feeling like he is being haranged by someone who is whining and defending himself against some crime, and trying to get out of the blame for it, but the blame here is the accusation of madness, not murder. He openly confesses to murder; it is the assumption of madness that bothers him, and he spends his entire tale trying to defend himself against that charge. Kind-of an interesting character, and he certainly makes the story more interesting. I hope that helps. Good luck!