How do I hook up my Gamecube to my Desktop?

Solution 1:

The cable you bought it meant to work in the other direction. It's supposed to convert a VGA video signal output from a PC into composite (or S-Video) video output that you can connect to a TV. In other words it would allow you to connect your desktop PC to a TV.

You need a converter that will convert composite video (or maybe S-Video if you have an S-Video cable for your GameCube) to VGA, HDMI or DVI depending on what inputs your monitor has available.