I love that you are being asked this question about Edgar Allan Poe's "The Tell-Tale Heart." While important, the focus of the story tends to fall on the narrator's unreliable nature and his obsession to the eye itself. The focus does not tend to center upon the play upon words, "eye" and "I," within the text.
Essentially, the connection between the "eye" and "I" in the text lies within the Old Man. The Old Man's eye is what the narrator has a problem with: "I think it was his eye! yes, it was this!" He, the narrator, hated the Old Man's eye so much that he wished to rid both himself and the Old Man of it. Unfortunately, the only way to rid them both of the eye was to kill the Old Man: "I made up my mind to take the life of the old man, and thus rid myself of the eye forever."
The narrator, determined to rid himself of the eye, took eight nights to take the life of the Old Man. On this night, the narrator was successful at entering into the Old Man's bedroom, pulled the mattress over the Old Man, and takes the Old Man's life. "His eye would trouble me [the narrator] no more."
The "eye" of the Old Man could not be removed from him without taking the "I" of the Old Man. What this means is that the identity, or being, of the Old Man lay in his life. Without his life, the Old Man has no identity. The two, the "eye" and the "I," are inseparable.