The narrator is a first-person one, the protagonist of the story, who is clearly mentally unstable.  This also makes him an unreliable narrator, one a reader cannot really trust to tell him/her the entire truth.  

In this story, the narrator reveals his powers of concentration through two methods:  1) by telling the reader what he has done, and 2) boasting about his meticulous nature and his cleverness.  He observes the the old man's eye that he is obsessed with and repulsed by night after night, standing perfectly still for hours upon end.  How anyone could stand still and peep through a crack in the door to stare at an eye for hours is beyond comprehension.  

In addition, the narrator has taken great care to clean up after the murder and dismember the body in order to cover his crime.  He is quite proud of his tidiness.  His confidence (arrogance, really) about his own ability to get away with this murder allows him to concentrate on doing the job the way he wants it done.  

The narrator is obsessed with the old man's eye and has made the decision to kill him. Rather than charge into the old man's room with a knife or attack him during breakfast one morning, the narrator carefully plans out his crime. For seven days, he slowly opens the door to the old man's room (taking an hour to get it open in order to stick his head in) and then lets just a thing ray of light fall upon the man's face. He also shows restraint and concentration in not killing the man when the eye isn't open.

On the eight night (he waited eight days to commit a crime he had already decided to commit) the eye is finally open and the narrator kills the old man. He also plans how to dismember the body, keep the area clean and how to hide the body.