Building a PC is really easy so you shouldn't be too worried about that. The first time I did it I watched this video while doing it since I was pretty nervous.
http://www.youtube.com/watch?v=d_56kyib-Ls
Anyway for a gaming PC, since you are on a budget don't spend too much on the processor. It will almost never be the bottleneck in modern games.
I play most of my games on ultra on a 1080p screen and can even play battlefield 3 and dirt 3 across three 1080p screens on ultra with this computer.
Radeon 7850
Athlon II x3 450 processor
8 gigs of Ram
Some generic shitty AsRock motherboard
Cooler Master 500Watt power supply
I build this computer a while ago and only upgraded the graphics card so far and I only had a $400 budget when I built it. So you could do a lot better.
From the website you posted with your budget I would probably go with the following.
AM3+ x6 FX-6300 $145 (CPU)
WD Black SATA3 1TB $92 (HardDrive)
AsRock 970-Pro3 $89 (Motherboard)
2G R9-270X Sapphire-OC $229 (GPU)
8G Kit 1600 Patriot-S $89 (RAM)
Antec HCG-Modular620M $125 (Power Supply)
SATA DVD RW LG $19 (Optic Drive)
And whatever case you like. Just makesure it isn't too small for the motherboard though thats pretty unlikely.
Total price will be around $840, excluding a windows license.
If you prefer Nvidia I would go with the 2G GTX 760 Gigabyte OC for $314. But your total price would become a little out of your price range.
