I'd look for a 3.5" HDD as oppose to a 2.5" as your case does not have a 2.5" bay unless you buy an adapter for an extra few dollars.
I also agree with HiTechObsessed about the RAM which is only 4GB, look for 2x 4gb's sticks which you can dual channel.
Another thing I picked up on is that your motherboard doesn't support crossfire so therefore if you are thinking about adding another card in the future, you will also have to get a new motherboard.
Finally, there is a possibility that the processor may bottleneck your graphics card in the first build. Perhaps go for an 8150 instead.
The second build is a bit better but keep in mind that the GTX 550ti is not a very powerful card and you may want to look into getting something better for playing Battlefield on ultra as this card will only be able to handle it on high.
Here is a build which I suggest. For the small budget, I tried my best and it should just about be able to handle Battlefield on ultra at a steady 30 FPS:
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ TigerDirect)
Motherboard: Gigabyte GA-78LMT-S2 Micro ATX AM3+ Motherboard ($47.49 @ Newegg)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory ($87.32 @ NCIX US)
Storage: Seagate Barracuda 500GB 3.5" 7200RPM Internal Hard Drive ($49.99 @ Amazon)
Video Card: Sapphire Radeon HD 7850 2GB Video Card ($153.98 @ SuperBiiz)
Case: Antec One ATX Mid Tower Case ($29.99 @ NCIX US)
Power Supply: Corsair CX 430W 80+ Bronze Certified Semi-Modular ATX Power Supply ($33.98 @ Newegg)
Total: $512.74
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-01-02 14:46 EST-0500)
Slightly over budget but it should give you some good performance especially for the amount of money paid. I have both the motherboard and processor in this build, both of which I highly recommend.
Hope I helped and good luck with your build!