A game computer...lets see here. Do you want to know the relative price of the components in a game computer or do you want to know a few pieces of general hardware that aren't that high up in the price range?
$750 is probably your best bet to get started. Afterwards you can upgrade it.
Motherboard - $150 - $200
Hard Drive - $80
Case - $100 - $200
RAM - $80 - $150
Power Supply - $90 - $150
Operating System - $100 +(unless open source)
Processor - $150 - $300
DVD Drive - $30 - $50
$750 is a slightly below-average gaming rig. You could do $500, but only if you were doing something like WoW and other lower power MMOs. It depends ENTIRELY on what you want to play, and how you want to play it.
Thanks for B.a. man,
be prepared to alter the build as you need, better psu/graphics etc but as boiler said, something like that will more than do you for Wow etc,
have fun and enjoy the build mate,