The story itself does not answer this question very clearly, so you are left to infer, or to make an educated guess, based on the information that we are given, and the historical context. There a couple possibilities:
1. They are father and son. This could be the case, but it is interesting that the narrator specifically mentions that there is no other motive than the eye. If they were father and son, then there would be a lot more emotional baggage there that might have given motive. Money also could have been a factor in a motive if they were father and son; perhaps the son wanted his legacy. However, the narrator insists that the old man had done him no harm, and that there was no motive. Granted, the narrator is unreliable, so we can't really take him at his word. He's a bit insane, and trying to paint himself in a good light.
2. They are both lodgers at a hotel, or at a boarding house or apartment complex. During Poe's time period, single men often lodged in boarding houses where they rented rooms in a house. So, maybe the old man and the narrator both lived in the same rental house. In such houses, rooms were individual and the bathrooms and dining areas were shared. It was an affordable and practical way of living for many people that lived in the city and had to work but couldn't afford a house. This theory would more accurately support the fact that he had no grudge against the old man, because they wouldn't have known each other well, and would only see each other in the halls and at mealtimes.
3. The old man is a lodger in his house. This is like scenario #2, but, the narrator is the owner of the house, and the old man is paying him rent.
Those would be my best guesses for the relationship between the narrator and the old man--either relatives, or, fellow-lodgeres in a rental situation. I hope that those thoughts helped; good luck!
I have one more possible scenario to add to the question that you raised.
My thought was that the young man actually is a lodger in the old man's home. He may be there as a type of assistant to the old man. The man talks about not wanting the old man's gold so maybe he is a relative that could stand to inherit from the man or maybe he kows where the man keeps things.
Poe is very careful to leave the reader wondering the nature of the relationship between the two men as well as the reason they live together. Even when the man talks about pulling up the floor boards, there is no slip of identification such "I pulled up my floor boards."
"I took up three planks from the flooring of the chamber, and deposited all between the scantlings."
I have always read it to mean that the narrator was a manservant to the old man. He brings him breakfast, doesn't refer to him as any sort of relation, and the police treat him as somewhat of an underling upon arrival.