This narrator is never fully characterized. Poe never gives us his name or tells what his relationship to the old man was. The two men are close enough that the narrator sees him every day for the week before he kills him, greeting him heartily every morning; and they are close enough that the narrator can intrude into the old man's bedroom nightly without having to sneak into the house. It is possible they are members of the same family, or that the narrator is the old man's servant, but readers never learn. They know only that he is passionate, unbalanced, and—if they trust his story—a murderer.
Like the narrator the old man is incompletely characterized—intentionally so. The narrator mentions he has gold and that he has an unnatural and filmy blue eye like a vulture's, but neither he nor Poe mention the old man's name. The old man is a passive character. He is rich and seems to have some authority, but he does little in the story besides sit in bed, open his eye, and cry out.
The neighbor, who never actually appears in the story, hears the old man shriek in the night. Suspecting foul play the neighbor contacts the police to lodge a report.
Three police officers come to investigate the report of a scream. The officers, who appear in the final few paragraphs of the story, are not differentiated and don't speak. Thanks to the narrator's calm and welcoming manner, they are at first convinced of his innocence, or so he says. Eventually, however, the narrator confesses to them.