Right now this is Windows-only
NOTE: In capture methods involving emulators, you should configure your computer to disable offloading.
Also note that while these captures are okay, they do not encompass all the traffic a Wii generates. For example, DNS lookups are not captured using this method. Captures from an actual Wii might have more information.
Fire up Dolphin and Microsoft Network Monitor.
Before you go to the wireless area of the game you want to capture, select "New Capture" then the "Start" button in Microsoft Network Manager.
Play online and capture some packets!
To export your capture, first click the "Stop" button in Microsoft Network Manager and select the Dolphin entry in the My Traffic list. Once selected, go the File -> Save as, then in the dialog that opens, make sure "Displayed frames" is selected under Frame selection.
Since we have more control with Dolphin, we can actually dump unencrypted SSL data without having to trick any servers. Below is a build of Dolphin that dumps unencrypted SSL traffic to two files, ssl_read.bin and ssl_write.bin. Saving these files along with your packet captures would help a lot, since we can look at how the encrypted and plain-text traffic interacts.
If you are going to use these files, make sure you delete or clear them between different capture sessions so old data doesn't get mixed in with new captures.
Dolphin Built with SSL Dumping