Build gaming PC rate the build?

I'm gonna be building a PC soon for games and I'm thinking of probably playing Battle Field 4 on it some other games I will definitely play are Skyrim, Chivalry, Minecraft, Loadout, Team Fortress 2. So I've chosen some pieces but I'm not sure which are better than others I'm on a budget of £590 (Pounds) For everything including monitor, keyboard and OS. So far I've built one PC but I'm not sure whether it's that great or not. How would it perform playing Battlefield 4? And Skyrim? On that "built" pc I haven't chosen RAM. Please say how well the "built" PC would be cos it is great for my budget. And "make" one as well with the parts and I'll check if it's in my budget. Please if you are gonna make a PC make one without any incompatibility's.

PC Built:
CPU: AMD FX-6300 3.5 GHz 6-core processor
Motherboard: Gigabyte GA-970A-DS3P ATX AM3+ motherboard
Memory: (Undecided on which one out of the two is the kingston one any good? It is cheaper than the patriot one.)
Memory 1: Kingston 12GB (3 x 4GB) DDR3-1333 Memory
Memory 2: Patriot Signature 8GB (2 x 4GB) DDR3-1333 Memory
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM internal hard drive
Video Card: Gigabyte Radeon R9 270 2GB Video Card
Case: Cooler Master K280 ATX Mid Tower Case
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply
Os: Windows 8 (OEM) (64-bit)

Parts:
CPU:
1- AMD FX-6300 3.5 GHz 6-core processor
2- AMD Athlon II X4 750K CPU - (Black) (Quad Core, 3.40GHz, 4MB, 100W, Socket FM2, AMD PowerNow! Technology)
3- AMD FX6300 Black Edition 6 Core (3.5/4.1GHz, 8MB Level 3 Cache, 6MB Level 2 Cache, Socket AM3+, 95W, Retail Boxed)

MotherBoard:
1- Gigabyte GA-970A-DS3P ATX AM3+ motherboard
2- AsRock FM2A55M-DGS Motherboard (Socket FM2, AMD A55 FCH, DDR3, 6 x S-ATA/600, Micro ATX, PCI-Express 2.0, AMD Radeon HD 7000)
3- Asus M5A97 LE R2.0 Motherboard (AMD 970/SB950, DDR3, 6 x S-ATA 600, ATX, PCI-Express 2.0, USB 3.0, Socket AM3+)

RAM Memory:
1- Kingston 12GB (3 x 4GB) DDR3-1333 Memory
2- Patriot Signature 8GB (2 x 4GB) DDR3-1333 Memory
3- G.Skill 8GB (2x 4GB) Dual Channel Ares Series Memory Kit (DDR3 1333, 9-9-9-24, 1.5v, Intel XMP Extreme Memory Profile Ready)
4- Corsair CMX8GX3M2B1600C9 XMS3 8GB (2x4GB) DDR3 1600 Mhz CL9 Performance Desktop Memory Kit

Storage:
1- Western Digital Caviar Blue 1TB 3.5" 7200RPM internal hard drive

Video Card:
1- Gigabyte Radeon R9 270 2GB Video Card
2- Gigabyte NVIDIA GTX660 2GB DDR5 PCI-E Graphics Card
3- Palit 2GB GDDR5 Nvidia GeForce GTX 650Ti Boost Graphics Card (PCI Express 3.0, HDMI, DVI-I, DVI-D, Display Port, 192-Bit, NVIDIA 3D Vision Ready)

Case:
1- Cooler Master K280 ATX Mid Tower Case
2- Inwin Fanqua Midi Tower Mesh Computer Gaming Case - High Performance Enthusiast PC Chassis is perfect for gamers, or any other user looking for a strong chassis with lots of smart features - No Psu
3- BitFenix Merc Beta Midi Tower Chassis - Black

Power Supply:
1- EVGA 500W 80+ Bronze Certified ATX Power Supply
2- Corsair Builder Series CXM 500W Modular 80 PLUS Bronze Certified ATX/EPS PSU
3- Zalman ZM500-GS 500W ATX Power Supply (Active PFC, 12v v2.3, Two-Forward Switching Circuit Design, Ultra Quiet 120mm Fan)

That's all. Now about a CPU cooler should I get one? Is there a good cheap one you know of? Thanks!

Added (1). Will an Arctic Cooling Freezer 7 Pro Rev.2 45.0 CFM Fluid Dynamic Bearing CPU Cooler work with my built computer? And is it any good?

That build looks solid friend, id also recommend you get a cpu cooler because the stock one can barely handle the 6300 on its own(I'm using it right now) and will constantly be at 50C at idle(unless you have the fans at 100% speed). As for the RAM, don't worey about the brand. If it does matter to you, get Kingston because they are well reputable and you should get the 12gb so when you upgrade in the future, you won't need to buy more ram.EDIT: ALSO for the cooler, I recommend the Cooler Master 212 EVO $35 on newegg.