All these methods assume you have modded your vWii. If you have not, follow this guide here.
The Wii U can natively run GameCube games with the help of Nintendont due to its hardware backwards compatibility with the Wii. However, the Wii U is unable to read GameCube discs. There are still multiple ways to run GameCube Games, but a modded Wii (or GameCube) is required to dump the discs.
Dumping GameCube games:
To dump GameCube games from a modded Wii follow this guide here.
GameCube games
Developer(s)
FIX94
Console
Wii
Version
4.431+ (December 18, 2016; 7 years ago (2016-12-18))
Nintendont allows you to load backups of your GameCube games natively on (v)Wii from an external drive connected via your Wii U's USB ports, or SD card.
Instructions
The process for setting up Nintendont on vWii is identical to that of the Wii's. Please follow the Wiki Hacks Guide page linked below.
UWUVCI allows you to inject any GameCube backup into a preexisting Wii Virtual Console title, package it with Nintendont, and install it to your Wii U's internal storage, or an external drive connected via the Wii U's USB ports.