tough order. playing on HIGH you will probably need a video card in the $200-250 zone, not leaving much budget for other parts. memory is ridiculous expensive right now, most kits being in the $70 range unless you can find some on a ridiculous good sale. also, if you have to pay for an operating system, thats a minimum of $100 of budget right there.
unfortunately the only way you're going to make budget on this and be able to run BF4 at high is going with AMD parts. and by cheaping out on parts that should be of high quality, like your ram, PSU, case, etc. I usually suggest intel, as i feel they are ahead of the game on intergrated features, calculations per clock, power efficiency, etc. but if $600 is absolute max, heres a suggestion that heavily favors dedicated graphics card over all the other parts. and oh yeah, if you have to pay for an operating system, its $100 over budget. all parts are from newegg.com
Processor: AMD FX 6300 - $120
Motherboard: MSI 970A-G46 AM3+ Motherboard- $70 - $10 Rebate
Memory: G.SKILL Ripjaws 4gb (2x2gb) DDR3 1600 Cas 7 model F3-12800CL7D-4GBRM- $48
Storage: Seagate Barracuda ST500DM002 500gb 7200 rpm 16mb cache SATA 6.0Gb/s HDD - $55
Power Supply: Thermaltake SMART Series SP-550PCBUS 550watt 80 PLUS BRONZE PSU- $50 - $10 rebate
Case: CFI Prime 223 CFI-A5396 AXT Mid tower- $40
Graphics Card:SAPPHIRE DUAL-X 100365BF4L Radeon R9 270 2GB 256-Bit GDDR5 PCI Express 3.0 CrossFireX Support Video Card With Boost & OC Battlefield 4 Edition- $220 + $8 shipping
Any CD Burner: $20
Total after rebates and shipping: $611, does not include tax.
This is probably about as close as it gets to $600 budget with it not being made of pure cheap garbage parts. if you can find better prices for other stuff, feel free to adjust your cart but if you keep anything in this cart, stick with the graphics card, processor, and at least the same chipset motherboard. 760g chipset motherboards that support FX processors are available for about another $20-$40 cheaper, but i would NOT recommend the 760g chipset, as its feature set very old and while it supports FX chips, will require a bios update for it and wont be ideal for a 6 core processor. Stick with 970 chipset if possible. Also, to allow room in budget for the graphics, i only put a 4gb kit in the cart, as u can upgrade to 8 later when you have more budget, and 4 will get you by for games that are currently out. but soon, we're gonna see some games requiring 6gb of system ram.
Good luck! hope you didnt need an operating system!
actually, ssddx's post is almost identical to this and has the best prices for quality parts, but all from diff locations. if u dont mind ordering off multiple sites, his suggestion is quite good.