You don't need a computer to connect your Wii to the internet. Probably the simplest way for you to connect, if your cable modem box is near the Wii is to get the Nintendo Lan adapter for the Wii. You just plug it into on of the usb ports on the back of the Wii and run a Cat-5 (ethernet) cable from the Wii to the cable modem.
On the Wii go to the setup menu and drill down to "Internet Access". Set up a Wired internet connection and you're set to go. With a wired connection there are no WEP codes to enter or SSID's to find so it's actually a much easier, more secure set up.
I looked up the set up for the gaming router you've bought and, again, you don't need a computer to set it up. Just plug it into the back of the cable modem and plug the power supply in. That sets up a non-secured wireless network which the Wii should be able to detect and connect to it. Your only 'catch' is that without a computer you won't be able to change any of the default settings.
Hope this helps.