I would say pick up a GTX 670 (or 570 if you need to save money), an i5 2500K, 16GB of ram, a motherboard to run those things, and a Corsair PSU at 650 watts. Since the i5 and mobo are older tech they'll be cheaper. You also don't need the fastest RAM but you'll want 16 for gaming. Some online games eat memory like you wouldn't believe (BF3 is particularly bad).
Then case and HDD. Get a case you like and as for HDDs... The SSD can wait to be an upgrade in the future. Just get a Western Digital Caviar Black 1TB, it'll work fine for gaming. I don't have high opinions of optical drives and I would say leave it out of the build.
That should drop you in around 900-1k depending on how diligent you are about finding sales. Do you need a monitor, keyboard, mouse, OS, and speakers as well? Because if so you're going to have to make some cuts. I assume you have those/ can find them in a free pile. You can always upgrade peripherals with time. Windows 8 is cheap right now and works perfectly well for gaming.
Look for bundles/bare-bones kits. They'll be cheaper than buying parts separate. Also remember that many stores will price-match to online prices so if you can find something cheap online with expensive shipping it's worth going to a real store and asking if they'll match the price.
Get a comfortable laser mouse, with a clicky scroll-wheel and side buttons. You'll thank me later.
I like NVIDIA graphics cards, they're generally better-quality than ATI as far as I can tell. Plus you get the cool CUDA stuff if you ever plan on doing any actual work with the machine. No one I know has ever gotten ATI's CrossFire to work. In fact, you don't even need more than one graphics card.
Use a decent-sized SSD (around 100GB or so should do) for your OS and apps, and use larger platter hard-drives for storage. If you want to spoil yourself and spend some more cash, you could get two SSDs and RAID-0 them for massive speed...
For RAM, usually you'd go for somewhat above whatever your games' recommended specs say.