Do you need keyboard, mouse, game controller, speakers and copy of Windows in that budget? Or, is $600 just for the PC hardware itself?
Short answer is $600 gets you an i3 paired with a GTX 960 or R9 380. Solid 1080P gaming. Case and rest of parts are kind of personal preference.
If you are willing to bump that up $200-300, you could get an i5 and GTX 970 which would go a long way to future proof your build.