You can take an APU (CPU + GPU), it have great performances like the AMD R7's graphics cards, and better than iGPU of intel
APU : AMD A10 7850K (25 ~ 30 fps in BF4 medium full HD) quad core 3.7 Ghz + 512 cores GPU (170 $)
Motherboard : MSI A78M E45 (65 $)
Storage : Segeate Barracuda 1 TB HDD 7200.14 (53 $)
RAM : Gskill 2 x 4 GB (80 $)
Case : Cooler master n400 (60 $)
Cooling : Cooler Master Hyper 212 evo (30 $ Optional)
PSU : Corsair CX430 V2 (40 $)
Total : 498 $ all from Amazon
Dont go for the APU build
-fx 6300 way over 7850k cpu
-r7 260x way over 7850k's gpu even if dual graphics is enabled
Now about the first build some fixes
-the 600w is more than you need really. You could buy same model from evga but on 500w
-the 260x is good gpu but it will perform the same in a fx6300 and athlon x4 750 if you get my point. http://www.tomshardware.com/reviews/build-your-own-budget-amd-pc,3807-2.html
-if you can buy a r9 270/x for 40-70£ more , than you have reason enoughs to stay with the fx6300.
-the 500w PSU wil suffice both cases.
The 270x is a real win really its for 1080p on high/ultra. 260x is fo 720ultra and 1080 med/high.
Try here http://makegamingcomputers.yolasite.com/ i use the 500 dollar build and play every game on high settings.
Both of those PSUs in the 400 and 500 build are trash. Seriously fire hazards they are so bad. Replace it ASAP and please do not recommend this to anyone. Also, neither of those is great for the money. Low quality parts and unbalanced builds.