Most TVs will have at least one auxillary input on the back, or sometimes on the front, of the TV. You're looking for a set of yellow (video), red (right-channel audio) and white (left-channel audio) plugs here. Older TVs may only have two connectors - one for video one for (mono) sound. In this case, just use the white audio cable from the Game Cube.
Once connected, you'll need to tell the TV to use that auxillary input. This is different for each TV unfortunately. Some TVs have an "AUX" button on their remote, while some put it somewhere in the channel rotation (try going to channel 02, and hitting the channel down button.) If the TV has multiple inputs, the remote usually has a "video" button to scroll through them, or once again, they may be at the end of the normal channel tuner.
However, if the TV is REALLY old (20+ years) then it may not even have any auxillary inputs, and just the connector for the TV cable. In this case you'll have to go to the store and buy an RFU adapter. You can either get one from the video game accessory area, or over in the TV accessories where it's billed as a way to connect a DVD player to "any TV". Replace "DVD Player" with "Game Cube" and just follow the instructions.
That should take care of 90% of the TVs out there. If you're still having problems, post what make and model the TV is (it's usually printed on the back of the TV)