Exemplifying Poe's signature style of arabesque, "The Tell-Tale Heart" is a narrative of horror and psychological torment in its many twists and turns and returns of narrative. Interestingly, some critics suggest that the old man is a doppelganger for the narrator, and the narrator's loathing for the man represents his own self-hatred. The narrator's obsession with the old man's eye seems to indicate his self-hatred as he projects these vile feelings onto something outside himself, a condition known as transference.
The eye of the old-man is identified by the narrator as a curse upon him. So, if he can rid himself of the eye--for he tells the reader that he loved the old man, and when the man's eyes are closed, he will not kill him--he will be all right. Thus, Poe with his patterns of arabesque "teases out the latent horror" of his image, a material image that forms an instance of a deranged and perverse aethetic as the narrator shares the horror with the old man:
Presently I heard a slight groan, and I knew it was the groan of mortal terror. It was not a groan of pain or of grief. Oh, no! It was the low stifled sound that arises from the bottom of the soul when overcharged with awe. I knew the sound well....
It was open, wide, wide open, and I grew furious as I gazed upon it. I saw it with perfect distinctness--all ..with a hideous veil over it that chilled the very marrow in my bones.
The psychological changes in the narrator parallel the physical events. The death of the old man, rather than ridding the narrator of a detestable object and relieving him, tortures the narrator since he has killed his doppelganger and thus, effects his own ruination. This removal of the perverse aesthetic destroys the narrator and he confesses in his desperate attempt to join with the old man in his self-hatred:
Oh God! what could I do?...They were making a mockery of my horror!...I felt that I must scream or die!
You have asked a very general question which needs to be narrowed down to indicate which specific aspect of this excellent story you want to analyse. Certainly one entrance point into this story is considering how the point of view is related to the action. One of Poe's defining stylistic features that he uses with great effect is the unreliable narrator - consider how he uses the unreliable narrator in other short stories as well as this one such as "The Cask of Amontillado" and "The Black Cat."
"The Tell-Tale Heart" is thus narrated by a first person narrator who, from the very first paragraph, we have our doubts about. One aspect that makes us suspicious about what he says and his mental state is the way that he continually protests that he is not mad:
True! - nervous - very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses - not destroyed - not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily - how calmly I can tell you the whole story.
This very first paragraph reveals beyond question that we are being told this story by an unreliable narrator because, in spite of his protestations of the last sentence, the narration that follows is anything but "healthy" and "calm" as we are given an insight into the madness of the narrator when he hears the beating of the old man's heart after he has been killed. Note too the way that he talks about his "acute hearing" also suggests his madness, saying that he heard "all things" in heaven and earth, and lots of things in hell besides.
Thus analysing the point of view of this classic story is one way into gaining an appreciation of the tale that lingers in the mind so long after you have finished reading it.