Sorry, the cheapest video editing software for Windows or Mac that I can think of that isn't horrible is about $100. The best open source stuff is all free, but all of those applications run on Linux, sometimes Mac, with no foreseeable ports to Windows anytime soon. If you ask around though, someone might have an old copy of something they don't use sitting around.
If I could choose anything, I'd probably go with Adobe Premiere Pro. I used the CS2 version once for a video project. Unfortunately, it's $800 more than you can probably spend on a small project like this, although with a student discount it's $350.
Premiere Elements has the needed multi-track editing functionality for a little under $100, but the dumbed-down user interface actually gets in the way. I get the feeling they may have dumbed it down even more since I bought a copy of version 4.0, so I wouldn't buy it without first downloading and trying the 30-day trial to make sure it's usable.
You might just have to export the audio from the game footage into Audacity, mix the voices in there, and then import it into Windows Movie Maker. That way, you get more control over the sound, but getting everything sync'd with the video becomes harder.