Alright. Thanks for posting, I'd love to have a say in this.
First off, here's a list I made, of parts you can get to build your own computer:
PCPartPicker part list /
Price breakdown by merchant /
BenchmarksCPU: AMD FX-8320 3.5GHz 8-Core Processor ($164.99 @ NCIX)
Motherboard: Asus M5A97 LE R2.0 ATX AM3+ Motherboard ($69.99 @ Canada Computers)
Memory: Patriot Viper 3 8GB (2 x 4GB) DDR3-1866 Memory ($64.30 @ Amazon Canada)
Video Card: MSI Radeon R9 280X 3GB Video Card ($330.35 @ TigerDirect Canada)
Case: Corsair 200R ATX Mid Tower Case ($53.13 @ DirectCanada)
Power Supply: Antec High Current Gamer 620W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply ($49.99 @ Canada Computers)
Total: $732.75
(Prices include shipping, taxes, and discounts when available.)(Generated by PCPartPicker 2013-11-04 22:30 EST-0500)(all parts are of course purchasable through various internet/irl retailers that sell in Canada)
This parts list is strongly assuming a few things. The first couple of things is that you're going to reuse parts of your current computer, mainly the harddrive, optical drive, monitor, mouse, keyboard. It is also assuming that your budget is around ~$750 at the max.
Allow me to explain the parts decisions.
CPU: Better than the ones you've mentioned above, and has 8-cores. Much horsepower for price, and will benefit you more than the other CPUs you've listed. The only minor problem that you'll see, is that the heat sensor function on some of the AMD FX processors doesn't function properly (unless it's hot/working hard, or so I'm told). So unless you're going to be monitoring temps, this isn't a big issue, and I don't find it to be a bit issue even though I sometimes monitor temps.
Mobo: I own this mobo. It's good.
RAM: ddr3-1866 speed, with 8gb and heat spreaders. Not much to say about this.
Videocard: This will strongly out perform the 7750 Diamond. This card will definitely suite your needs, and will have very little problems touching max settings on games such as BF4, Bioshock Infinite (play with the light settings though. Most videocards have trouble with lighting, since it consumes so much gpu power), the Metro games, you name it, and it will more than likely play the game at max settings (with varying FPS based on the game). Since BF4 is the game you want to play this will hold through BF4 no problem.
Case: this part is up for debate. If you like the case your computer has now, then reuse your current case. Otherwise, this case is cheap, well-liked and it has decent looks.
Power Supply: It'll have enough power to run your system, so you won't need to worry about anything. It's semi-modular as well, so it can help keep cord clutter to a minimum in your case, which will lower your temps. If you don't know yet, keeping temps low increases the life span of computers, meaning they'll hold onto power for longer.
Again, a lot of these parts are up for debate. The CPU, RAM and Mobo are to ensure performance, so I would strongly recommend you follow these parts up. Other than that, the videocard can be changed to one that is cheaper by about $75 or so, and will still be capable of running BF4 on max settings. It won't get quite as many FPS, and it won't hold up to future games quite as well as the current GPU I have in the parts list, but it will work just as efficiently. This is entirely up to you, though. The case is also up for debate, as there are some cheaper options, although cheaper cases generally have less room for cable management, are poorly constructed and have less airflow. Again, entirely up to you whether or not you want to choose a cheaper case in the interest of saving money, or if you want to pick a different case so that you can have different looks.
A tip of advice with computers though, is get the powerful parts and spend the extra money. This will mean that the parts will last longer, perform better and hold up better. You'll save money on computers in the long run, and it will benefit you more than ever.
Now, to get into detours.
If you want a new harddrive and just ditch the old one, you can get
this one, and this will increase the cost by said amount on product page. This may also mean that you will need to reinstall your OS. If you don't have the CD and product code on your current computer available to reuse, then that will add an additional $80-100ish. So it's strongly recommended that you keep your current harddrive, unless it's an old dinosaur and on the verge of dying.
If you want to keep your processor extra cool, you can look into getting an aftermarket cpu heatsink.
this one is by far the best for the price, and will perform perfectly. I use it as well. Definitely recommend this if you're looking at getting an aftermarket cooler.
If you want a cheaper videocard, go with
this one, and it will reduce the price on the build by $100, while still handling BF4 at max settings. I already addressed the issue of videocards, so I don't need to repeat myself.
More cooling needed/wanted? You can get a few
120mm fans for cheap ($4-5 per fan depending on what fans you get. The one I linked to doesn't have to come in the green LED, haha. There are also other options, but this has the best performance/quiet ratio for its price). The case supports up to 8, but if you want to get more, 1 or 2 would work perfectly fine. Any more and you'll need to get a fan controller, because the mobo won't handle that many fans.
Anything else I missed?
EDIT: Oh, and considering this will strongly out-perform anything for the same price on the used market you're looking at, it's definitely recommendable that you just build your own computer. You'd have to wait for quite some time and get really lucky to come across a computer that would perform as well as the one with the parts I've linked to you for less the price on the used market.