If you wish to play anything GameCube on the Wii, much like GameBoy Advance games on the DS or DS games on the 3DS, all Wii-related functions will shut down, resulting in everything Wii related not working.
You should go out and buy a controller and memory card (However, beware, there are a bunch of Chinese knock-off controllers out there on eBay. They work alright, but they break down easily after a few months), or if you really want to, you could purchase it on the Wii's Virtual Console for 1,000 Nintendo Points ($10USD, which is honestly cheaper than a GameCube controller and memory card). If you purchase it on the Virtual Console, you can use your Classic Controller since it emulates the game, rather than going into straight-up GameCube mode.