I all of a sudden noticed my GC making strange noises when I have a disc in and playing. Like you can hear it processing the game or something.
Is this normal and I just wasn't aware or is it a sign that my system is going buh-bye?

Depending on what game you are playing and how it accesses the disc (not to mention how often) could very well vary the sound of the drive reading mechanism. Gamecubes are built tough but they do break (though 90% of them that come into the store I go to are physically caused.. i.e.: kids). If I'm playing by myself I can hear it access the disc as well so I'm pretty sure it's nothing to worry about. If you're really concerned compare it to another Gamecube with the same disc and if it's different, go from there.

Unless the sound is distractingly loud, I wouldn't worry about it. Most consoles make some kind of sound when they're reading a disc. The Dreamcast was one of the loudest.