Go to the Libretro main site: https://www.libretro.com
Click the "Downloads" link in the top menu to go to the current release version (as of this writing, the current version is 1.3.6).
Navigate down the current version directory to the "nintendo" subfolder, then "wii", and download the "RetroArch.7z" file.
Create a folder called "apps" in the root of your SD card, and make a folder called "RetroArch" inside that.
Extract the contents of the downloaded file into the RetroArch folder you just made on the SD card.
The .7z extension is a 7-Zip archive. This can be downloaded for free from the 7-Zip site: http://www.7-zip.org
Create a folder to store your ROMs on the SD card. The name and location don't strictly matter, but the simplest option is a folder called something like "roms" in the root of the SD card. Put the randomized ROM you generated earlier into this folder.
Put the SD card back in your Wii, and load the Homebrew Channel from the main menu.
RetroArch should appear in the main menu of the Homebrew Channel once it's loaded. Select it to load the emulator.
Once on the main menu of RetroArch, select "Load Core" and scroll down and select the "snes9x-next" core. This is a modified snes9x core for libretro.
Once the core is loaded and you're back on the main menu, select "Load Content", navigate to the folder where you stored the ROM, and select it to load.
That's it! You can mess around with the video and audio settings in RetroArch if you like.
RECOMMENDED: By default, there is a sizable audio delay in the emulator settings. If you go to Settings -> Audio, you can change the audio delay from the default of 64ms down to a minimum of 8ms. Turn this all the way down. On PC, this would normally cause audio crackling if the buffer is too small, but the smallest size works fine on the Wii.
OPTIONAL: You can get a SNES to GameCube converter to play with your actual SNES controller: http://www.raphnet-tech.com/products/snes_to_wii/index.php