Suggestions on a build for a gaming computer $600 budget

Hi, first I would like to tell you that i think it is really hard to build a "decent"computer for 600$. Even if it were 600 € it would still be hard.I would recommend you to buy a cheap case that already has a PSU to minimise costs like this http://www.amazon.com/Cooler-Master-Elite-431-Plus/dp/B005ZCTJ9Q/ref=lh_ni_t?ie=UTF8&psc=1&smid=ATVPDKIKX0DER. I would also recomend you to buy an i5 if you want to game but its seems hard to get an i5 with your budget so I would recomend you to get an AMD CPU coz they are cheaper. I think an FX 6300 6 core would be good for you (http://shop.amd.com/us/All/Detail/Processor/FD6300WMHKBOX). You will also need a cheap motherboard so that you can get a good gpu, I would recommend you to get a ASUS M5A78L-M LX3 as it is really cheap (http://www.amazon.com/M5A78L-M-LX-PLUS-Micro-Motherboard/dp/B005WUUFBW). For RAM 4 Gb is enough for gaming i would recommend g.skill 2X2 at 1600 Mhz (http://www.amazon.com/G-SKILL-Ripjaws-240-Pin-Desktop-F3-12800CL9D-4GBRL/dp/B002RPCLH6/ref=sr_sp-btf_title_1_4?s=electronics&ie=UTF8&qid=1379602081&sr=1-4&keywords=G.SKILL+4GB+DDR3+1600MHz+CL9+Ripjaws+PC12800). For the GPU, if you want to be able to run all recent games I only recommend the gtx 760 (or some amd equivalent, honestly I am not interested in AMD GPUs so I can't say anything about this) the ASUS version of this would be the best as it is the one with the lower temps (http://www.amazon.com/GTX760-DC2OC-2GD5-GeForce-256-bit-PCI-Express-OC-selected/dp/B00DBPKEOI/ref=lh_ni_t?ie=UTF8&psc=1&smid=A1TSN35NIS24VZ).
The only 2 problems with this build is that you will be very limited by your motherboard and you will also be generating a large amount of heat inside the case because of the GPU.
However I think that the build that I have done does not pass your 600 $ budget so you can save a bit of money after you buy your computer and buy some decent air flow for your case or just water cool your CPU so that all the heat produced by your CPU goes directly out of the case.
I would also like to hear your opinion, good luck!
By the way I am so dumb that I forgot that the case I mentioned didn't had a PSU, here is one http://www.amazon.com/gp/product/B001G0WPLK/ref=ox_sc_act_title_1?ie=UTF8&psc=1&smid=ATVPDKIKX0DER