Do you also need an OS? Putting a gaming computer together with a monitor and OS for $600 is nearly impossible. Here's a build without an OS for just over $600:
ASUS VH222H Black 21.5" 5ms HDMI Widescreen 16:9 Full HD 1080P LCD Monitor Built in Speakers 300 cd/m2 1000:1 (ASCR20000:1) w/ SPDIF out - Retail
http://www.newegg.com/Product/Prod [...] 6824236053 $159.99 - $10.00 MIR
Total - $624.90 - $10 MIR
Windows 7 will add $100 to the build unless you're a student. You can get it for $30 if you're a student.