Now, honestly, there are a couple of different approaches that you can take that all work just fine, but I kind of like the top-down way of most important functionality first the best, so I'll try to talk you through that. It's what I did when I, a couple of years ago when I still had money, wanted to do but never did when I bought a laptop instead.
You begin by thinking about your gaming needs, i.e. whether you'll be playing the most taxing games or not, and decide on the price range for your GPU based on that. You need more expensive hardware to run the graphically most intense games. That's pretty much the most important thing for a good gaming PC to have. Find your way to Tom's Hardware and start looking for the most recent articles on those things, preferably this particular one since it's the newest, and do a quick comparison of the benchmarks the ones they recommend have managed to hit. Personally, I'd say you're probably going to want to go with an enthusiast class graphics card since your budget is limited. Maybe GTX 760/770, if you're willing to pay $250/$315 for it.
Then, start looking for a good processor to go with it. I'm an Intel guy who last kept up with stuff when the Ivy-Bridge models were coming up, so I can't really recommend that many things at the moment, but look for their best to second-best 3.2-3.4GHz processors. Unless you want to get something else, I don't know. Here's the relevant article for CPUs. Also, while you're deciding on things, try to find out about how much energy each component will gobble up. You'll need that information for later when you come back to pick a PSU.
Then, after those two are checks, start looking for RAM, 8 to 12 Gigs at least, 16 if you can find that for cheap. That's relatively easy since you only pretty much have to find something that seems good and doesn't have a super eccentric input. Look for a big enough hard drive as well. Preferably two: a smaller SSD that you will only install your operating system and the other essentials on to make the beast run smoother, and a large normal internal hard drive. Once that is done, check all your existing components and check very carefully what sockets your motherboard is going to need to have. You want to figure out the exact number when you start looking for one to give life to your system. Most recommended gaming motherboards have all the needed sockets by default, but not every single one, so you will want to pay some attention to what you're getting. After you find your mobo, run a few calculations to see what kind of a Power Supply you're going to need, and find one. A 600-650W PSU will probably do.
When all that is done, you just need to find a pretty casing that will fit everything in it, and get a good cooling system to keep it from overheating and losing maybe even as much as five to ten years of its lifespan. That's a lot in a poor person's money. Probably a fan or two for the GPU and maybe for the processor, plus a water-cooling system if you want to be sure.
Then put it all together, and ta-da, you have your new gaming rig. I don't think your budget will afford you a really good monitor on top of all that, so go with something that has a good enough framerate (120hz) and that has gotten okay reviews, but that isn't too expensive. Peripherals are all on you. I'm not going to start giving advice on that because it's more a matter of personal preference than anything else.
I think that's it. I hope I didn't forget anything that is important. You could also make it easier for you by using a readymade PC part picker that has everything you need listed. Really, I could have just posted a link to that, but I wanted to write so I hope you'll forgive me.