Just so happened I was looking on here and saw your thread. As it happens I'm involved with a group of guys that play BF3 every night on the PC and rent their own servers to play on. Your right about BF3 in particular being a better experience on the PC.
Your budget is pretty bang on the money and you have 2 choices.
Build the system yourself or...
Configure the system and get someone else to build it like, CyberPowerSystems, Chillblast, Palicomp, PCSpecialist or Dino PC. All of which carry a good name, some are better than other in terms of value, service etc so its a good starting point to see whats out there. Might be worth getting back up to speed with the latest gear out there as well. Pick up the odd mag here and there. Computer Shopper, PC Advisor, Custom PC (definately) or PC Format (again definately as its more of a gaming orientated mag). The website HardwareHeaven.com is always a good source as are many others, Kitguru, OverClockers Club etc.
Intel are offering the best CPU's these days by and large, AMD just don't have the grunt and use a fair bit more power to achieve the same results as Intel if their overclocked. Intel boards are Socket 1156 these days with some good products on offer from Asus and MSI. Worth checking their websites to get a feel for the products on offer.
Graphics I would stick to a budget of somewhere near £200.00 to get the most out of a game from either Nvidia (can't go wrong) or AMD (some good bang for buck). Stick to a single card though...none of this silly crossfire (two cards) stuff as you would need a bigger PSU to power it.
PSU's, I would stick to either Corsair, Antec, CoolerMaster or BeQuiet in that order. They all carry a good name. Some near 500 to 750 Watt would probably meet you needs. Ebuyer.com is a great site to look at prices and what is on offer.
Case's I would go for something with good cooling. Looks wise is always a personal taste. You can see my case down below. I've found it strikes a good balance between cooling, price and most important...size to accommodate all the the gear that goes into a case these days.
Here is an example of what I have...just an example as its about a year old.
Manufacturer: Self-Built With My Own Two Hands
Processor: Intel Sandybridge i5-2500K (overclocked to 4.5GHz)
Memory: Corsair Vengeance 8GB 1600MHz DDR3
Hard Drive: 1TB Hitachi (2x500GB) 7200RPM
Video Card: MSI GTX570 Twin Frozr III Power Edition OC
Monitor: LG 22" W2261VP
Sound Card: Asus Xonar DX
Speakers Headphones: Turtle Beach Z6A 5.1
Keyboard: Steelseries MERC
Mouse: CM Storm Spawn
Mouse Surface: CM Storm Tactics Pad
Operation System: Windows 7 Home Premium (64-bit)
Motherboard: Asrock P67 Extreme 4
Computer Case: Cooler Master HAF 912
PSU: CoolerMaster Silent Pro 700 Watt PSU
You can more expensive but I don't see the point. There is a price point sweet spot at the minute and you budget definately hits that. I wouldn't as you say buy cheap, spend twice, not recommended.
You can feel free to swing by our forums at www.solo-elite.com
on the tech section to ask some gamers what gear they would recommend. Friendly guys and the ones who play BF3 like I mentioned. We do play a good range of popular titles from time to time and been gaming for about 13 years online as a group.
Feel free to drop by but I'll keep an eye on this thread.