Nowadays, with barebone packs, mail in rebates, and other sales, you could probably get away with a great computer for 800 dollars. If you're not one of those people who are into brand-hype and looks, purchase a simple aluminum (it's less noisy) case and if need be, drill your own side mounts for fans. This shouldn't cost you much.
A normal barebone kit will cost around 200$ and normally includes a CPU, motherboard, and power supply. Memory and hard drives are at an all time low, and their prices continue to fall.
Obviously other parts and accessories will vary greatly. If you're a big gamer, processing speed, RAM, and your video card will be factors you want to consider.
It all depends on your maximum budget, and whether or not you're patient enough to buy parts one little bit at a time as they go on sale.