The first obvious problem is that killing people is morally wrong. The arguments the narrator puts forward justifying his decision show just how mentally disturbed and disconnected from reality he actually is. In many ways, the record of the narrator's steps in making the decision form a perfect record of the disintegration of his mind.
The narrator tells us from the outset his reasons for the murder:
I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire. I think it was his eye! Yes, it was this! One of his eyes resembled that 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 for ever.
The first problem we see here is that the narrator thinks there is something evil or malevolent about the old man's eye, which was pale blue and covered with a film. A sane person would simply note that one should not kill someone for having blue eyes. The film on the eye, which seems mysterious and ominous to the insane narrator, is actually an ordinary effect of aging.
For the narrator, the next problem was that he he wanted to kill the old man in his sleep but while his eye was open, because in his disturbed mental state, he believed wanted to kill the eye, not the old man himself. This caused a problem, because the old man, as most people, did not sleep with his eye open. Finally, he solves the problem by casting a narrow beam of light on the eye, leaving the rest of the old man in darkness. Next ensues a second problem, a worry that the neighbors might hear him (which, in fact, they do, leading the police to visit).
His third problem is hiding the body, which is solved by placing it under the floorboards. The final problem is that the narrator is haunted by guilt, and imagines that he can hear the dead man's heart beating, which the narrator solves by revealing the body's hiding place to the police.