The AMD chips are genuinely a nice bargain for what you pay, but they are more-so designed for multi-threading processes (like audio/video/photo production and editing). Yes, some there are more titles coming out that support multi-threading, but most of the currently popular titles (like BF4) don't seem to really benefit from it; at least, not from what I've been reading (no personal experience). The FX-6300 is basically a triple core CPU with multi-threading, so it would excel primarily in places where two extra logical cores are useful (not necessarily useful with all games, from what I've read/heard).
So, I'll compare the FX-6300 to the i3-4150 (a fair comparison):
If what I've been reading is fairly accurate, and BF4 does not really supporting multi-threading, then the 6300's extra core's single thread processing power will not be noticeable, when compared to the 4150. Even if the multi-threading is supported, the most noticeable difference will come from overclocking the FX chip. At that point, you're no longer saving money, and you're then going to spend a little extra to OC your FX chip. Also, AMD parts run hotter, and consume more power... so there's always a give and take. I've also heard that the stock CPU coolers that come with the FX chips are better off replaced, regardless of OC'ing. If that's the case, you're still spending more. The Intel stock CPU coolers are just fine for your stock clock speeds. With all of this in mind, I would automatically go for the i3-4150.
This build sheet is my personal recommendation, based on prices reported
today. Some parts, like RAM, GPU, and MoBo could be switched out for cost reasons when the time comes to buy. At that point, I'm sure we'd all be happy to help if you have questions. For CPU, HDD, PSU, and case, I would definitely recommend sticking with the listed parts:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($98.99 @ NCIX US)
Motherboard: MSI Z87-G41 PC Mate ATX LGA1150 Motherboard ($82.99 @ Amazon)
Memory: G.Skill Sniper 8GB (2 x 4GB) DDR3-1600 Memory ($72.00 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($53.99 @ Amazon)
Video Card: Asus GeForce GTX 750 Ti 2GB Video Card ($139.99 @ NCIX US)
Case: Corsair 200R ATX Mid Tower Case ($54.99 @ TigerDirect)
Power Supply: Corsair CX 430W 80+ Bronze Certified Semi-Modular ATX Power Supply ($49.99 @ Amazon)
Total: $552.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-10-01 18:42 EDT-0400
Now, keep in mind that the 750 Ti is likely to drop in price soon (NVIDIA is supposed to launch some new cards in the very near future), so keep an eye on the 750 Ti; that will be your best bet for an NVIDIA card for cost/performance ratio. If it doesn't drop to a price that's within budget, I would get the standard 750, which should be suitable enough; and it already comes within budget (give or take $5). If you go the AMD route for video cards, do note that the drivers are not designed to work with Intel chips equally, so you're likely going to see less stable performance. If you had a more expensive build, you might not care as much about this; but with your restrictive budget, you want as much stability as you can get.
It's worth noting that this build has an estimated power consumption of around 215w, so don't be tempted to get a 500w PSU for the sake of overpowering. Larger power supplies are designed to be efficient with heavier current draws, so you will see less efficiency from a stronger PSU than you would with one that's rated in the 350-450w range. Try to size the power supply to the build, and it will keep you from potentially overspending, as well as being less efficient. If you find a 500+w PSU that's significantly cheaper ($10+ less) than a quality 350-450w, then go for it; otherwise, the minor savings won't ever benefit you outside of the upfront cost. This especially goes for 80+ Bronze rated PSUs.