One thing you can do is visit your local computer stores and see about having them build you one. Most will do it, especially the small mom and pop stores.
The broadstrokes to consider is that laptops are always bad ideas for gaming. They cost more, give you less perfomance, and are much harder, if not impossible, to ever upgrade. So buy a desktop.
The other things I would say is try for a box with 8gig of ram (should cost about $50, I recently got 16gb for $100), you probably will not need more. For example, Skyrim is not going to use more than 4gb.
Given that you are on a budget, I suggest something with an Intel i3 processor. Just look for the best you can get for the money you have.
Graphics cards are the last major concern. A vanilla store-bought PC is going to have an unimpressive graphics card. So if you go that route, leave aside a few hundred dollars to buy a better card. Something like http://www.bestbuy.com/site/EVGA+-+GeForce+GTX+660+2GB+GDDR5+PCI+Express+3.0+Graphics+Card/6820043.p;jsessionid=3EED3937574C8A5936394B4C326FD2A5.bbolsp-app06-29?id=1218808244387&skuId=6820043&st=geforce&cp=1&lp=1 if you have a lot to spend, or maybe a http://www.bestbuy.com/site/NVIDIA+-+GeForce+GTX+550+Ti+1GB+GDDR5+PCI+Express+2.0+Graphics+Card/2375089.p;jsessionid=3EED3937574C8A5936394B4C326FD2A5.bbolsp-app06-29?id=1218323853938&skuId=2375089&st=geforce&cp=1&lp=6 if you have less.
If you go the vanilla computer route, do not fear. Even http://www.bestbuy.com/site/HP+-+Desktop+-+6GB+Memory+-+1TB+Hard+Drive/6927728.p?id=1218814316101&skuId=6927728 would do you good if you added one of those two cards above to it. 6gb is a little light on ram, but you could spend an extra $50 and double that.
If you are having a box built, then the last big thing to consider is the power supply. Do not skimp on this, as everything else needs a good PSU, especially your graphics card. Look for something with at least 800 watts, if not more. They are not going to be cheap, but it is worth it. Skimp on your hard drive or dvd drive instead, use an old monitor, keyboard and mouse for a while, but do not go cheap on your power.