I included a copy of Windows 7 because I didn't know if you would need it. This should run minecraft much better than you need, 100-300 fps on max settings probably. If you're doing any recording you can still expect over 50fps on max. If you don't need the Windows 7 disk than you can take it out and save $85, making it about $367 total. You could even play Battlefield 4 on medium settings at 50 - 60fps
Wait, where is the graphics card in that build? Don't know much about minecraft's system requirements but I am positive that BF4 wont run max without a GPU.
So I doubt you could do 50 FPS medium on BF4 but couldn't find benchmarks for it.
I'm not an expert, just thought that 50 FPS BF4 sounded a bit out there for an APU, but it should be enough for your original question which is minecraft at 30 fps, medium settings.
Edit: Read on a BF4 forum that a guy was getting 20-30 fps if lucky with the A10-5800k and 8 GB DDR3 RAM
http://battlelog.medalofhonor.com/bf4/forum/threadview/2955064767486258048/
Personally i would recommend a discrete GPU but im not great at picking out parts yet (dont have much experience picking mobos or CPUs), but someone could probably recommend a good solution in your price range.
I notice that for the PSU you have Micro center as the source. If Micro center is an option for the OP he will be able to get an A10 Kaveri 7850K and motherboard combo cheaper than the APU / MB combo price in the best answer build above. The A10 7850 bundled with an Asus A88MX MB combo will be right at $175 after taxes. Slight gain in the APU with the benefit of Mantle support and stronger graphics on the chip for no additional cost.
Back to the OP's question, an APU will do just fine for minecraft. Minecraft's performance is really only limited by the CPU and GPU has little effect. From my little experience I noticed no change in minecrafts performance after switching from a HD 6670 to an R9 280x which is a huge leap. All new AMD APU's will be sufficient to run minecraft and in the case of the A10-7850k and its bros with mantle can even run bf4 on low, maybe mid settings without a discrete GPU