One way is to softmod or hardmod your GameCube and run a BIOS that has video mode forcing capabilities. Another way is to use a PAL -> NTSC signal convertors and a Freeloader/Action Replay. I've also heard of using component video cables instead of A/V and RF will work, but I wouldn't know anything about that.
I suggest getting your GC modded in some way and either run GCOS, Ninjamod, or Qoob Pro BIOS to get it working. GCOS and Ninjamod will work with both softmods and hardmods whereas Qoob Pro is a modchip. It's a matter of preference really. I suggest just getting a Qoob Pro and flasing it with a BIOS ver1.3 or later to play PAL games on an NTSC system.
I am not sure this tip will work, but it is worth a try:
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Change screen refresh rate
While powering on the system, hold B until the message "Would you like to switch to 50hz mode?" appears. This allows you to switch the screen refresh to match that of PAL games. If playing on a PAL system, the message will be "Would you like to switch to 60hz mode?", allowing the screen refresh to match that of NTSC games.
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Thing is, not all PAL games have the option of changing the refresh rate. For example, Shadow the Hedgehog and Paper Mario 2 have that option, whereas Killer 7 doesn't. He'd also need a Freeloader to even boot the game at all, since GameCubes are region locked and won't boot a PAL game on an NTSC console. Also, all Freeloader does is make the GameCube region free. If you boot a PAL game on an NTSC game console expecting it to work with an NTSC television set, it's going to come up black and white or not at all. Unless that TV set supports the PAL standard, it won't properly display it.
It's much easier using a BIOS that has force video mode since it removes all the guess work. Granted, a few games won't work with it, but it's a hell of a lot easier and cost efficient than any other method. Not to mention it allows you to back up your original copies so you won't have to worry about damaging them.
