The narrator is clearly suffering a psychological breakdown in the story. I mean, he is relating how he murdered an old man for no reason other than his eye creeped him out, essentially. Since he tells the story in first-person, the reader cannot determine how much of what he says is true; thus, he is an unreliable narrator. He also repeatedly tells us that he is perfectly sane, & goes to great lengths to convince us of that. Yet he is clearly describing the actions of a madman. This is evident from first line of the story: ‘‘True—nervous—very, very dreadfully nervous I had been and am; but why will you say that I am mad?'' We quickly realize that the reason someone would say that is because he is insane.
As far as the alliteration in the title, it has a few different purposes. One is for remembering: phrases tend to stick in your head if they use alliteration. However, it also creates a rhythm to the title, almost like the beating of a heart. Read it with emphasis on the "T" sounds; you'll hear a beat very similar to what the narrator imagines coming through his floorboards.
In Edgar Allen Poe's story "The Tell Tale Heart" the man begins his story by trying to explain tot he reader that he is not crazy. He wants to convince the reader that the act of murdering the man with the "evil eye" occurred because it was necessary not an insane act. The man demonstrates that his mental condition is steadily deteriorating. Poe illustrates this process by the use of alliteration. He uses words in textual content and visually. For example:
"Hearken! and observe how healthily, how calmly, I can tell you the whole story.
In the line above, Poe places emphasis on the "c' sounding like a "k." (alliteration)
The use of alliteration tells a different story than the way in which the narrator wants the reader to think. The expression in the words indicates to the reader the opposite of what the words state. The man is demonstrating a forced calmness.