You have to use a SD Card. I think for Wii U you have to use a game exploit so things like Letterbomb are out. Smash Stack is the most popular since Smash Bros is an extremely common/popular game.
So you can just follow the directions for installing the Homebrew Channel on a Wii, and it will be the exact same on a Wii U in Wii mode. The GAF wiki is extremely thorough. Almost too thorough, IMO, it makes it seem more complicated than it really is.
Edit: And using Gecko or USB loader is really easy once you get them on there, and even with Gecko that is just a matter of putting it on the SD card. And you can create a
forwarder channel so you don't have to load up the Homebrew Channel every time you want to launch your region-free app of choice. That requires some extra effort but it's useful to have.