While "updating" your hardware using that method works, might I suggest an alternative plan?
1. Instead of replacing your monitor with a TV, get a new video card that has video-in capabilities. Having the RCA AV jacks, RF input, and whatnot is definatly recommended.
2. Instead of replacing your PC, get the "Gamecube PC add-on." It's pretty cheap, only $100 USD, and fits comfortably next to your PC that now has video-in on the video card. To hook it up, simply use the cables that came with the "GC PC add-on."
3. I have to agree with this step. Get the memory card. You'll be glad you did.
4. As an added bonus for getting the "GC PC add-on," you get a free controller! Mine works great, but I've bought more so that I can have some multiplayer sessions.
While I don't know the exact price for all of this, it shouldn't be to much. Maybe $150-$200, depending on the video-in device you get. Another suggestion that I'd propose, to help integrate the "GC PC add-on" better would be to get a USB adapter for the controller that came free. That way, you can use it on you PC, and on your new "GC PC add-on." I got one for $20. Works pretty good.
Anyway, hopefully you've found some useful suggestions on how to get GC games running at full speed using your PC (to some extent). Mouser X over and out.