There's also the option to build around the G3258 CPU. The right choice is about how you intend to game now and your upgrade plans.
It's an interesting choice:
a) In many games it beats the FX-6300, and
b) it's cheaper (which allows the difference to be put elsewhere), and
c) with the 1150 socket you have the option to throw in a 4-core Haswell CPU later.
FYI, Asus supports overclocking on H81, H85. There's quite a few links discussing building around this CPU.
The best choice depends on whether your games are quite CPU dependent. Skyrim and Starcraft 2 will run better on the G3258. I would imagine BF4 would run better on the FX-6300.
Other comments:
1) The NXZT 220 is a really nice case. While you are on a budget and it's $45 it also comes with TWO nice fans which is rare at this price. The 12cm would be moved to the front and I'm pretty certain they plug into the CPU_FAN slots on the motherboard so you can keep the computer fairly quiet.
http://pcpartpicker.com/part/nzxt-case-caso22001
2) You don't mention Windows, but I'll just say that WINDOWS 8.1 64-bit is the best option. Some people keep recommending Windows 7. Windows 8.1 is better in many ways, and the annoying interface (to me and most) is easily avoided with Start8. It's basically identical visual wise then.
3) HDD:
It's about $7 more for a 1TB drive. Aside from capacity, smaller drives get slower quicker. By that I mean data at the 500GB mark on the 1TB HDD is roughly 80% of full speed but 50% of full speed on a 500GB HDD (due to the angular velocity).
http://pcpartpicker.com/part/seagate-internal-hard-drive-st1000dm003
4) DDR3 memory:
While 8GB is optimal, for most games 4GB is sufficient, for example Skyrim maxes out at roughly 1.75GB and Windows can get by on 1GB or less. On a budget this tight I would personally stick with 4GB rather than take it from one of the few places you can like the VIDEO CARD.
However, I would make sure to get a motherboard with FOUR slots then add another (preferentially identical) kit later.
If in doubt, look up the specs before you buy but I've tested over 200 games for System RAM, Video RAM, and the impact of SSD vs HDD performance (surprise SSD isn't a big deal usually aside from load times).