In the second sentence, the narrator asserts that "you," the reader, "know the nature of [his] soul." What do we know about the narrator when he first makes this claim and what do you know about him by the end of the story?

We know very little about Montresor, the narrator, by the second sentence of the story. We do realize from the first sentence that he is a proud and vengeful man who vows to punish when he is insulted. This opening statement provides a hint that he is not a saintly and forgiving human being.
By the end of the story, we know that Montresor is a man capable of cold-bloodedly walling up his enemy in a cold, moldy catacomb and leaving him alone to die. This act indicates that Montresor's soul, from a spiritual point of view, is not in good order. He has committed a heinous sin, and now, fifty years later, he wants to talk about it. His statement near the end that "my heart grew sick" and his confession of what he did indicates that he does feel some remorse for his act.