Why does the narrator kill the old man?
This is a story in which the doppelganger, or double, is key. Poe has the narrator say a number of things that point to his identification with the old man. Even the apparently random 'eye' the narrator gives as his reason for the murder is a homonym of 'I', and so the narrator can in a real sense be seen to be killing the old man in an act of loathing directed at himself.
Although there is no specific reason given for the narrator's murdering of the old man, it seems as though there is some severe psychological problem present. Although it is just a guess, it sounds a little like paranoid schizophrenia.
For an unknown reason, the old man's cloudy, pale blue eye has incited madness in the narrator. Whenever the old man looks at him, his blood turns cold. Thus, he is determined to kill him to get rid of this curse.
This was all caused by the narrator's fear of the old man's cold blue eyes, it was not that he was mad or crazy as his cool and measured actions was not the type a madmen would behaved regularly. Every single night, he would approached the old man's house and observed him sleeping in his bed and in the morning, he would behaved in a way that nothing had happened and everything was back to normal, no trouble whatsoever. After a few weeks of observation, he thought randomly it was time to kill the old man, which was really out of a sudden and unexpected. This story was trying to to provide a detailed study and a analysis of paranoia and madness through a short story.