Building new $750 gaming PC - advice needed

here's my suggestion, working off newegg. The following system included a 1920x1080 monitor and is faster than the 4100 system you were looking at.

i5-2400: $189.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

AS Rock Z68M/USB 3: $89.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Patriot 8 GB DDR3-1600: $34.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Corsair CX500W PSU: $49.99 w/ Promo code. plus $20 rebate
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

HIS HD6870 w/ free DiRT 3, plus $20 rebate: $159.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Not as powerful as the 560-ti, but solid anyway, and enough cheaper to get the monitor

ACER 21.5" 1920x1080 monitor: $109.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Hitachi 500 GB 7200 rpm hard drive: $79.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

LG 22x DVD burner: $16.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Rosewill R102 case: $29.99
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

total: $761.91
shipping $22.99
Rebates: $40

$744.90 after shipping and rebates

Get rid of that FX CPU. Any FX CPU will bottleneck a growing number of game titles and the quad core variants aren't much better than a quad core Athlon II chip. I would put the quad core FX chips between the quad core Athlons and the quad core Phenoms, much closer to the Athlons. Since gaming tends to utilize highly parallel CPUs like the six and eight core FX CPUs extra cores pretty poorly more cores does not help much if at all, depending on the game being played. Phenom II quad core chips represent the best of AMD's gaming performance right now. They are pretty far behind Intel since they are three or four years old so yeah... not good for AMD.

Battlefield 3 is very CPU intensive and will be bottlenecked by pretty much any AMD CPU. The quad core Phenoms may be okay with it if you overclock one of them (Phenom II x4 955 BE or better) but otherwise you would be much better off with a recent Intel quad core chip, preferably the i5-2500K for overclocking and the i5-2400 for non-overclocking systems.

If you are upgrading your monitor to something capable of higher resolutions than 1920x1080 then you will be better off with a similarly (or better) performing video card with more than 1GB of video memory.

Due to your constrained budget I would not recommend this but I will recommend switching to a Radeon 6870. It has similar performance to the GTX 560 ti but can be found much cheaper than the GTX 560 ti.

Grab a Radeon 6870 for about $170 or less. You can also get the Cooler Master Hyper 212 CPU cooler instead of your choice for about $20. It will still take decent overclocks and can get the i5-2500K up to around 4.4GHz easily enough and maybe a little farther with a second fan installed for a push-pull configuration. I own this cooler and put a second fan I had on it and it works great for me. If you get the i5-2400 instead of the i5-2500K then you can either get the hyper 212 or just stick with the stock cooler that comes with the CPU.

If the monitor is not included in your budget you can switch that i5-2400 in the above poster's build with the i5-2500K and get a decent case because that case is kinda small.

the case is small, but it works fantastically. You're just limited to 9.25" or smaller video cards (which isn't that hard to do nowadays).

Can that Radeon 6870 fit in that? I don't know how large it is.

that 6870 from HIS is 9.0" long. In the same case, I have a 6850 that is 9.1" long, so
it will fit.

No problem then but I just don't like small cases like that... I like more room.

Personal preference though, size of case will not effect performance unless heat becomes a problem and I really don't think it will.