Here's what I'd recommend. You can check prices on the parts as I really don't want to dig for the prices.
CPU: X3 440
Mobo: Gigabyte GA-790XTA-UD4 (this might be expensive there, so I can find some others that are cheaper)
RAM: Any non-OCZ 2x2 GB 1600 mhz CAS Latency 7 kit
GPU: HD 5770
HDD: Samsung Spinpoint F3 500 GB or Seagate 7200.12 500 GB
PSU: Any 650W unit from Corsair, XFX, Silverstone, SeaSonic or Antec
Case: Antec 200, Antec 300, Antec 300 Illusion, Coolermaster Elite, Coolermaster Centurion 5, Coolermaster 690
Optical: The cheapest SATA DVD burner you can find
HSF: Coolermaster Hyper 212 Plus or Scythe Mugen 2 Rev. B (SCMG-2100). This can wait.
I'm positive that'll be over budget, but it might now.
EDIT: Crap. Just noticed you need an OS and a monitor. You're not going to get a quality build if you need those under the $600 budget. That'll add a good $250-300 by themselves, leaving only $300 for the tower. That makes it impossible to build anything that will game.