The Game Cube Linux project recently released a short tutorial to show how to patch and recompile a vanilla kernel to let it run on your Game Cube.
If you are going to recompile the kernel you must consider you are going to do it probably from a fully functional Linux environment: be prepared to cross-compile for the ppc32 architecture, cause that's what the Nintendo console is running on. You will find a link to a PowerPC compatible toolchain for x86 in the tutorial. The GC Linux projects provides you with some patches appliable to the latest vanilla kernel and even with a customized pre-made .config, so that you can avoid the configuration stage.

Ppcnerds first wrote about this project months ago, when it was in early alpha stage. Several steps forward were done during this range of time: one of this giant steps was the porting of the SDL library, that allowed developers to build and run succesfully mplayer, mame and a bunch of games (doom and tetris among them).

Linux is growing on gaming consoles and is deserving attentions from IBM: at this URL, in the Developer Works section of Big Blue's site, a small section dedicated to Linux on Power gaming consoles has been added.

Linux on these consoles is just interesting, a possibility to expand the use of something thought just as a gaming platform but structured with good hardware that allows you to do much more. A multimedia box, nothing revolutionary, but an add-on to the gaming strategy. Sony is officially selling a Linux kit for its PS2 and several distributions are known to run on the Microsoft Xbox (have you ever thought of a Xbox cluster?).
On the Power side the Nintendo GC might be sooner or later in company of a rumored ppc970 Xbox, but these are just rumors as said.