From the outset, the narrator tells the audience that he is not crazy, yet in doing so he emphasizes the very thing he seeks to disavow. "But will you say that I am mad? The disease had sharpened my senses, not destroyed..." Later the narrator accuses the audiences, "You fancy me mad. Madmen know nothing," and the very assertion he seeks to disprove is reinforced. After setting his plan in motion, the narrator tries to convince the audience further. "And now have I not told you that what you mistake for madness is but over-acuteness of the senses?" The last example occurs just before the police show up when the narrator reassures the audience "If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body." At every turn the narrator reveals his madness even as he attempts to deny it.
Enotes has a great resource for this at the following link.