it's pretty easy to get a decent gaming computer for <$1k nowadays
assuming you're not building one:
get a decent prebuilt computer (preferably with a decent PSU) and add a GPU
you might want to look for a cheap quadcore build (the one above seems a tad pricey since it's dual core and refurb)
the one above might be okay for anything up to an ATI 6850/70, or an nvidia 460 or 560 TI
you might need to upgrade the PSU
i had weird chunking problems in GTA IV with lynnfield i7 860 + ati 5770, so a nvidia card might be safer
then (unless you already have them) you add a monitor, speakers, and maybe keyboard/mouse (although you might want to just play using an xbox 360 controller or something)
if you don't want to do any assembly at all you'll probably have to buy from ibuypower, sager/clevo or something