If the only time your computer is freezing is during game play, then you're problem may not be the hardware.
Look into the software, Ubuntu has been know to be a little buggy when it comes to the latest games.
Update Ubuntu, the drivers, Steam, and Games.
I found an update for driver, I will install later and see what happens. I tried Steam OS too and the behavior is the same (guess is exact same driver).
Its first time that use Radeon GPU. I had another 3 PCs with Linux, AMD CPUs, but all with NVidia GPUs and never had problem like this.