Just make sure there isn't a clause in your agreement with your ISP that states you are only paying for 1 IP address, as your independent PC will have its own IP and everything on the router will have another if I am not mistaken. If they notice, and you are only allowed to have one, they may try and charge you more money.
Why wouldn't you run the computer off the router's switch as well? If you are trying to keep the other computers on a separate network put a switch behind the router instead of a hub in front of it