Brawl Friend Codes (not to be confused with general Wii Friend Codes) are found by first selecting the Wifi option from the game's main menu and (by hook or by crook) getting the internet connection to work right. Rom there, just select "With Friends" and then "Friend Roster."
From there, you can see your current avatar, player name (up to five letters) and most importantly, your own FC. When you connect for the first time you are asked to type in a name for yourself, so maybe you also see it then.
Anyway, my code is 4725-7617-2850, and I tend to go by "PIXEL."
If people want to exchange stuff like snapshots, replays and custom stages, they have to trade general Wii codes as well as Brawl codes. Those codes can be found in the mailbox section of the Wii Menu. From there, hit the second of three icons on the left, the Address Book. Then hit the large middle option and you should see your own 16-digit "Wii Number" as well as the option to put in others.
I wish I could be more active here, but I'm too busy drawing furry for Concession to draw scaly here...
Thread for Comissions (Nothing to show for it yet, but I doodle every day to build confidence)