Hey there mate. 600 bucks is no problem, i have worked with people on a 150 dollar budget before!
Now, i would also reccomend an AMD processor, purely because of the fact that you can get so much value out of so little cash, So obviously we will be going with an AMD chipset motherboard. Your motherboard:
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131651 - $90
Feature rich motherboard with the latest technologies such as USB3.0
Now, because this is a gaming pc you will want a processor that can pack a punch, but wont break your budget. Your CPU(Processor):
http://www.newegg.com/Product/Product.aspx?Item=N82E16819103924 - $150
an AM3 processor running at a very high speed of 3.6Ghz !
Cases are one of those things that you can just get amazing value for money. Your case:
http://www.newegg.com/Product/Product.aspx?Item=N82E16811103030 - $50
A quality budget case with two led fans, and room for upgradeability in the future
Hard Drives are a little harder to get bang for your buck, but i think i have made a pretty good choice here. Your HDD (Hard Disk Drive/Hard Drive):
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136784 -$120
500GB leaves room for lots of data in the future. Comes with a 2 Years Parts and Labor warranty
Memory is so cheap these days! Your memory:
http://www.newegg.com/Product/Product.aspx?Item=N82E16820104206 - $25
Its a dream come true for you! You can get yourself 4GB of high performance RAM from one of the leading manufacturers out there, with LIFETIME PARTS WARRANTY!
A trusty power supply is essential, luckily there are many cheap options from leading value brands. Your power supply:
http://www.newegg.com/Product/Product.aspx?Item=N82E16817182044 - $40
Rosewill make quality value products, and the 500watts in this psu will easily be able to feed your machine with the power it needs
This is probably the most important part of your machine, as it is a gaming computer. So i have saved 100 dollars for a quality HD graphics cards which should run most of the games on the market today! Your Graphics Card:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150577 - $100
This is a great card using a quality chipset made by amd. HD ready, very stylish design by XFX.
Currently we are at $565 thus we have 35 bucks left to spare.
The system will run perfectly with those components but i highly reccomend getting an optical drive (DVD/CD player) in your system unless you are going to be downloading every game online. So here is your Optical Drive:
http://www.newegg.com/Product/Product.aspx?Item=N82E16827106289 - $18
Runs and burns every type of DVD/CD on the markey today except for blu-ray
I hope you liked my reccomendations. Your budget was $600 and i came in at $583 with 17 bucks to spare.