most people use Fraps to record their game. You have it running while playing and it will record what you see in video files.
The files are really huge (in original resolution with not much compression), so you need to compress them afterwards; a bit of editing doesn't hurt either. You can do both in the Windows Movie Maker; you can get it free from Microsoft and I think it offers the basic functions you need. Other programs are Magix and Sony's Vegas.
After you're done editing, you can save/export your video; I think WMM only lets you create Windows Media Video format, but I guess that's fine for the start.
Then, you have your final video file which you can upload to YouTube.
I have a newer computer but not much beyond that. I'd like to record a campaign runthrough but I've never done that. I'd appreciate it if you could tell me what I need to do so(PM me the details). What video recorder do use, microphone and anything else you think is relevant. Thanks