First you need to forget about your computer for a moment.
On your wii and go into settings (click on 'wii' at the bottom of the main menu) then go into wii settings.
Scroll accross until you get to 'internet' and click.
Then you need to go into settings and get it to search for an access point, and click wireless, not wired if it asks you at some point.
Once your wii has searched, it will show you all the wireless networks it has found. If there is one there that you recognise, connect to that. If it asks for a password, then its the one you use for your computer to get onto the internet.
If there is a network there you dont recognise, but is not password protected, then you could access it, but you would technically be using someone elses internet connection, and that isnt very polite!
If there is no wireless point found, then your router is just a wired type and you will need to replace it with a wireless router. (which you can still connect your computer to with a wired connection if you want)