I'm not sure the narrator started out with a plan, but this is how the plan unfolded:
The narrator admits he was overly nice to the old man in the week before he killed him, in order to allay suspicion, "I was never kinder to the old man than during the whole week before I killed him." The next step was that the narrator watched the old man every night while he was sleeping. He snuck into the old man's room just around midnight to try to sneak a glance at the eye. "And this I did for seven long nights - every night just at midnight - but I found the eye always closed; and so it was impossible to do the work; for it was not the old man who vexed me, but his Evil Eye."
The next step was that he acted perfectly normally during the day, speaking to the old man and asking him mundane questions so the old man would not suspect anything was amiss, "So you see he would have been a very profound old man, indeed, to suspect that every night, just at twelve, I looked in upon him while he slept."
Then the narrator snuck into his room on the eighth night and accidentally frightened the old man. He waited an hour, letting the terror build up in both the killer and the victim. Finally, the narrator snapped and carried out the deed. "With a loud yell, I threw open the lantern and leaped into the room. He shrieked once - once only. In an instant I dragged him to the floor, and pulled the heavy bed over him. I then smiled gaily, to find the deed so far done."
Poe's "The Tell-Tale Heart" is arguably one of his most disturbing. The action unfolds this way.
First, the narrator tells us that he believes the old man with whom he lives has an "evil eye." The eye is weirdly cloudy and sight of it drives him literally insane (or so he thinks is the cause.)
The narrator decides that the only way to be rid of the sight of the eye forever is to kill the old man. The narrator sneaks into the man's room at night, but cannot bring himself to murder him until he has one more look at the eye.
The old man, however, sleeps soundly. It is not until the eighth night that he awakens. Sensing someone in his room, he fears (rightly so, but not for the reasons he believes) that "Death" is approaching.
The narrator goes berserk. He thinks the loudness from terror of the old's heart will betray him. He carries out his hideous plan, but it goes afoul when the narrator believes the heart beats on.
He decides to dismember the man and bury him beneath the floorboards.
Alerted by the scream of the old man, neighbors *do* become wary and call the police. The narrator tries to rid them, but they do not leave. The longer they stay, the more agitated the narrator becomes. In his mind, the beating heart becomes so unbearably loud that he finally screams out his confession: "I admit the deed!—tear up the planks!—here, here!—it is the beating of his hideous heart!"