To what effect are sentence structure and punctuation used in the second paragraph of "The Tell-Tale Heart"?

This is a great question. Here is the passage in question:

It is impossible to say how first the idea entered my brain; but once conceived, it haunted me day and night. Object there was none. Passion there was none. I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire. I think it was his eye! yes, it was this! He had the eye of a vulture -- a pale blue eye, with a film over it. Whenever it fell upon me, my blood ran cold; and so by degrees -- very gradually -- I made up my mind to take the life of the old man, and thus rid myself of the eye forever.  

The person speaking is the murderer, someone obviously insane. So, the problem for Poe is, how does an insane person write? Or, more precisely, how does an insane person who is trying to rationalize his actions write? If we look at the sentence structure, and the rhythm of this paragraph, we can learn a thing or two. 

Take the first sentence, which joins two ideas: the "impossibility" of how the idea of killing the old man came to him, and the consuming passion the idea exerted over him. Neither statement makes any rational sense, but the construction of the sentence, which joins two independent clauses with the phrase "but once concieved" suggests a kind of rational cause and effect which belies the actual content.

Or take the next six sentences, each very short. In this passage, Poe is building a rhythm similar to the ravings we might expect from a deranged killer. Note that each sentence is an argument against killing the man -- he is testifying to the irrationality of his act. When he finally breaks that rhythm -- "I think it was his eye!" -- his statement makes no sense at all, but the way he says it -- the exclamation mark, the tacked on "yes, it was this!" (suggesting the speaker has fixed on the reason only at that moment) -- has a kind of authenticity to it. The speaker might be lying or mentally unstable, but the manner of his delivery has the ring of truth. In other words, he sounds crazy.

There is nothing about the eye that would drive most people to murder; the speaker describes it as "the eye of a vulture -- pale blue, with a film over it" -- as if this would explain his reaction. The dash suggests equivalence: "vulture eyes" equal "pale blue with a film" -- but this itself makes no sense. What vulture has blue eyes?

Whatever the reason, the sight of the eye made his "blood run cold." In a construction that mirrors the beginning sentence of the passage, the speaker uses another phrase -- "and so by degrees -- very gradually" -- to connect the cause (his blood running cold) with the effect ("I made up my mind to take the life of the old man"). As with the first sentence, the logic suggested by the syntax is completely at odds with the content. The sentence structure suggests a cause and effect relationship that simply does not exist in any real way.