Is there some way to hook up your Gamecube or Wii to your computer?

I don't think so

To connect these to your computer you need a video capture card installed on your computer. However, these introduce too much of lag to the video signal making them unsuitable for gaming.

So your best bet is to find a monitor with a VGA input and get one of those Composite-Video to VGA converters (NOT AN ADAPTER!) and connect directly to the monitor.

The closest option for this would be to purchase a video capture card / device (like Elgato's Game Capture HD series) or even use a TV Tuner device (like Hauppauge's WinTV series) to get the console's A/V signals into your computer to use with a Broadcasting / Recording program like OBS or XSplit OR with other programs that accept A/V signals (like Google Hangouts).

Please be aware that this only allows you to broadcast the A/V signals & doesn't permit remote input. If you want to do remote input WITHOUT using an emulator, you'll have to design your own remote rig & software to relay the inputs into the console.

Hope this helps!

If you have a capture card, you can set it up to work like a camera and mic to your system to be used any way you want.

This would include recording, streaming, or screen sharing through compatible services.

You would want something like this for SD inputs: http://amzn.to/2viKV1E