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. You do not need to mod the vWii to use injections.
Extract the zip, then copy the contents of the zip to the root of the SD card.
Move back to your console and Launch WUP Installer GX2 from either the homebrew launcher (Tiramisu) or the Wii U Menu(Aroma).
Install the forwarder to either the nand or a USB.
Once finished, exit to the Wii U menu and the forwarder should be there.
Launch the forwarder. You can now play your games the classic way!
The Nintendont forwarder only has SD card support. Having the games on a USB will not work due to the forwarder acting as a WiiVC inject, which doesn't have USB access.