You posted, "I have since hooked it (the router) up to my computer and it works like a charm?.
You created your own conceptual problem.
The Router is Not Hooked to the computer. The Router is hooked to the Modem, and the computer is hooked to the Router.
Wireless Routers usually have four ports for wired computer, and can provide connection to many Wireless computers.
Each computer connects to the Router on its own, so whether a computer or a game Box is connected or Not does not matter for the rest of the computers.
It is like the water in the house. It comes through a Main faucet (the Router) and from there goes to the water locations in the house. You do not need the water in the kitchen (computer) to be open and running in order to have water in the Bathroom (GameBox) and vice versa.