Actually you can get a pretty good gaming build for 600$. but I'd advise a budget limit of around 700$, It gives some better wiggle room.
You can't get too great for $600. A good motherboard, processor, and average GPU, 4GB RAM, average case. A bit more. Mostly average stuff.
Ok, then what can I get for $700/800?
For $800, you can get a pretty good one. For $700, you can get a good run. You might want to expand your budget another $100 or $200 from $600. It helps and may cost more, but really pays off. Wait for Soviet to come. He can put together a good build for you.