Installation of Homebrew on a Wii U requires a game disc. For the sake of this tutorial, we'll be using Super Smash Bros. Brawl, as it's probably the most popular game for which an exploit exists. Please note, though, that you can use LEGO Indiana Jones, Yu-Gi-Oh 5D's Wheelie Breakers, LEGO Batman, Lego Star Wars or Tales of Symphonia: Dawn of the New World-based exploits in a similar fashion.

Installing Newer Super Mario Bros. Wii on Wii U's Virtual Wii Mode.

Download Riivolution from HERE.

Locate* a "boot.elf" file in the package you've downloaded from above and put it on the root** of the SD Card.

* - To unpack a zip file, download a program like 7-Zip.
** - Look HERE for an explanation what "root" of an SD card is.

Download the Newer Super Mario Bros. Wii package from HERE and locate folders "NewerSMBW" and "riivolution". Put both of them on the root of the SD card. Here is a sample picture of what this will look like.

Now, download Smash Stack. For American Wiis, download is HERE and for European Wiis HERE and for Japanese ones HERE.
Note than in case of European Wiis, you have to choose an appropriate pack depending on if you have a Super Smash Bros. Brawl save in the System Memory.

Plug in your SD card and locate a folder named "private". If it exists, rename it to "private_old". Then, put the private folder from Smash Stack's pack onto root of the SD Card.

The SD card will look like this.

"NewerSMBW", "riivolution" and "private" folders, plus a "boot.elf" file.

Launch the Virtual Wii mode. Put in Super Smash Bros. Brawl and insert the SD card.

Load up Super Smash Bros. Brawl...

If asked to create a savefile, respond negatively.

Navigate to the Stage Builder found under "Vault".

The screen will turn black for a moment and riivolution will start. Quickly eject Super Smash Bros. Brawl and put in the New Super Mario Bros. Wii disc.

Newer is up and ready to be played.

Installing easy-access channels (NOT RECOMMENDED)

There are additional things to do to make loading Newer more convenient, but they're NOT RECOMMENDED on the Wii U, as Nintendo may decide to start going after people with hacked channels on their Virtual Wiis. If you do not care, feel free to continue.

You can click "Install" in the riivolution menu, which will install a Riivolution channel, for easy access.

You can also install the Homebrew Channel.

Create an "apps" folder on the root. In it, create a "riivolution" folder. Then, move the boot.elf from the root to it.

Download the HackMii Installer v1.2, and place the boot.elf found in the package on the root of the SD card.

Launch Super Smash Bros. Brawl and navigate to the Stage Builder.

The ? button will take a while to appear, but be patient, and then press it.

On this screen, press continue.

Here, press "Install the Homebrew Channel". After it finishes, click on "exit".

The Homebrew Channel should be installed and ready! Riivolution will be accessible from there.