I've successfully used this sort of cleaning goop stuff on a lot of user-interface electronic equipment; it's kind of like Gak but has more tensile strength, and is infused with disinfectants.
The basic idea is you squish the stuff into the device, and it fills the cavities with goop; when you pull the goop back out, it brings the crud with it while leaving fast-evaporating disinfectants behind.
From personal experience I know that it works fairly well in general on keyboards, mice, and PS3 controllers (the amount of junk it can pull out of a keyboard is impressive, but that's what it was designed for); from what I remember of the GC controller, I don't foresee see any particular problems as long as you're careful around the thumbsticks - you don't want to accidentally leave goop behind there.
The downsides are that the stuff is surprisingly expensive, and doesn't last as long as you might think; keep in mind that it basically eats all the junk it comes into contact with, so it maxes out at about three keyboards-worth of grime, at which point it kinda starts falling apart.
I only had it at the time because an off-brand version of the stuff was on sale at a local electronics store. If cost is an issue, you're better off with a bottle of isopropyl, a pack of q-tips and a paper towel, as @k1DBLITZ says; it's not as fun, but is definitely effective.