Hey Jason,
For character animation for games you have two main roles, either in game animation or cut scene animation. It'd be good to show a little of both. So Joojaa's point of actually having your animation in a real game engine would show off your understanding of in game animation. For in game you need to show you understand how to make things loopable and able to have things like having a run cycle seamlessly tie into a jump animation clip without there being any popping. But if you do do an actual real time game, I'd just screeen capture it to a dvd instead of giving the actual executable, most studios looking at reels just have a dvd player in front of them and having to install an executable just slows them down, plus they might be cautious of bring computer viruses into the studio. Or just fake an in game experience using Maya's trax editor, or something similar. And then in terms of showing cut scene animation, just show off good body mechanics and good acting. You can probably cover this in about 3 clips, have a character interacting with something heavy, have two character in a 10 second dialog (from your favorite movie), have two characters physically interacting (fighting). Just make sure to show off a good sense of timing (record yourself doing it for reference), a good sense of weight/balance, good arcs, subtle moving holds, strong silhouettes in poses, contrast of poses, thoughtful acting choices, etc. Don't worry about using somebody elses character rig, just make sure it's not copy righted. In editing your reel, show your best work first, and also something impressive to end on.
Hope this helps!
Jason