LOL, I guess by "I don´t want to record music that plays in the background through iTunes" he meant he wants to use his own background music, but not record it with iTunes.
If that's all he wants to do, there's a way to do it with Fraps recordings by using VirtualDub to demux the audio from the Fraps video, then mix it with his music track via Audacity, then add them back in via VDub.
Only thing is you need to make sure you don't change the length of the audio track, eg music track you add needs to be equal or shorter length.
I have used this method numerous times though, and VDub and Audacity are both good tools and free. Plus Audacity has effects you can add, like fade out, fade in. I'd often combine that with volume level to lower the volume of the background music during character dialog in the game. Works quite well, and easy to use.