The Wii is backwards compatible with GameCube games. As such, it is also able to play GameCube backups.
In order to load GameCube backups, you need an additional GameCube loader. There are various methods available; you will need one of the following:
It is recommended to use Nintendont, as it has high compatibility, is easy to install and works well with existing USB loaders.
Notes:
Loading from Disc: Loads from standard DVD or a mini-DVD. Multiple games can be put on the same DVD through the Multigame ISO creator.
Will not work on newer Wiis and WiiU due to drive incompatibility.
vWii Compatible: Works on the WiiU virtual Wii mode.
Doesn't use MIOS: Normally, GameCube games are loaded though the MIOS. These launchers use methods to bypass the MIOS.
Note that there is only 1 MIOS on the Wii, so only one MIOS-related launcher can be installed at the same time.
The launchers that don't use the MIOS can coexist with other launchers.
Memory card emulation: Ability to emulate GameCube memory cards through save files on SD card.
Audio Streaming: Only certain GameCube games use audio streaming. An incompatible loader means the game will play, but without sound.