In the opening paragraph, the narrator tries to convince his listener that he is not mad (crazy). He admits to having a disease but he claims that the disease has made his senses sharper. We have a narrator who claims to be reliable but his anxious insistence makes him seem crazy indeed. It therefore comes as no surprise when he says that he loves the old man but then endeavors to kill him.
The narrator says he loved the old man, the old man had never wronged him, and that he did not covet the old man's money ("gold"). Illustrating his mental instability, the narrator claims that it is the old man's eye that drives him to insane thoughts.
I think it was his eye! yes, it was this! He had the eye of a vulture—a pale blue eye, with a film over it. Whenever it fell upon me, my blood ran cold; and so by degrees—very gradually—I made up my mind to take the life of the old man, and thus rid myself of the eye forever.
From here, the narrator again tries to prove his sanity by describing how strategically he went about killing the old man. But what was it about the eye that drove him to murder?
The narrator claims that it is not the man himself who he has the problem with. It is only his eye, his "Evil eye." The narrator clearly has some insane, superstitious notion that the old man's eye is evil and is therefore some kind of threat to him. Approaching the old man, the narrator describes his unreasonable fear and terror of the eye:
It was open—wide, wide open—and I grew furious as I gazed upon it. I saw it with perfect distinctness—all a dull blue, with a hideous veil over it that chilled the very marrow in my bones; but I could see nothing else of the old man's face or person: for I had directed the ray as if by instinct, precisely upon the damned spot.
The narrator has a self-destructive impulse. If it is not suicide, it is the desire to commit the murder and then be caught. Indeed, this does occur. The guilt manifests when he hears the man's heart beating through the wall. Note the pun of "eye" and "I." In killing the "eye," the narrator symbolically wishes to destroy himself as "I." So, part (or perhaps all) of his desire to destroy the old man's "Evil eye" is actually a subconscious attempt to destroy himself.