I have a budget of $400-$500 to build the best budget gaming computer money will buy, and can unfortunately only use this website for computer parts: http://www.gamedude.com.au/
So far I have got this: http://localhostr.com/files/2Ubu5D4/2398u.jpg
I am currently thinking of stretching the budget to get a 7850 instead of a 7770 and maybe going from 4 to 8 gb of ram, but that will be costly. What do you guys think? Is there a better build than what I currently have?
You might be able to save a few dollars by getting a download version of Windows instead of a DVD and by getting perhaps a Seagate Barracuda HDD, which is typically cheaper than WD.
As for the GPU, yes the 7850 will have better performance, but I seriously doubt that you'd be able to run it using a 350W PSU. In fact, I'm not sure the system as you have it now can be run on such a small PSU. I'd recommend no less than 450 and better 500 or 550. You will also likely run into bottlenecking issues with a faster GPU using the Pentium.
The unfortunate fact is that it's very difficult to build a "gaming" PC for under $500 unless you already own many of the parts. You might consider looking into an AMD APU build - it's not a strong graphics card, but you can't beat the performance for super-budget builds.