If you're plugging in an ethernet cable, you're not using wifi as by-definition wifi is wireless.
You either use your Wii's built-in wifi along with your wireless router, or you pick up the RJ45 adapter which plugs into one of the USB ports on the back of the Wii, and then use a cable between your Wii and your router. Some people use a USB wifi adapter on their PC, but that setup is more-complex and requires your PC to be on, and if you're going to be spending that money you'd be better-off spending a bit more for a wireless router instead.