Since the merger of the Wii-Network branch in 3.5-2143, Dolphin can now use the Nintendo WiFi Connection to play Wii games online. Since it uses the official Nintendo platform for Wii online gameplay, Dolphin users can play online with not only other Dolphin players, but with real Wii players as well. For copyright reasons, some of the files required to access the Nintendo WiFi Connection cannot be distributed with Dolphin and have to be obtained by the user.
As of May 20th 2014, the Nintendo WiFi Connection has been terminated. This guide was written while the WFC still existed and no longer works. These instructions will remain up to help provide a reference point for users dealing with custom servers. More specific instructions may come once the custom server scene has settled.
Download NUS Downloader. This program allows you to obtain various system files directly from Nintendo's official servers. Extract the archive then run the program. Check the "Create Decrypted Contents(*.app)" box, then select the latest version of ISO 13 by clicking Database > IOS > 000000010000000D - IOS13 > Latest Version. Click the "Start NUS Download!" button to download the Wii operating system files from Nintendo.
Once the download has completed, go to the directory where you extracted NUS Downloader and browse to the titles\000000010000000D\1032 directory. Make sure that the 00000011.app file exists; you'll need it for the next step.
Download the extract_certs.exe program. Drag and drop the 00000011.app you got from NUS Downloader onto the .exe for the program, and you should see three files appear next to 0000001.app: clientca.pem, clientcakey.pem and rootca.pem.
If you are using Linux or OSX, compile extract_certs.cpp (in C++11 mode) and use this to dump the certificates.
Copy these three PEM files to the My Documents\Dolphin Emulator\Wii directory. After this is done, you should be able to access the Nintendo WiFi Connection with Dolphin! Enjoy.