For an inexpensive solution I would not consider the A6-3670k which is based on AMD's 1st generation APU (CPU + GPU). It is relatively inexpensive, but tI think you would be better off with the A10-5800k for $130. AMD just released the newer A10-6800k, but the performance increase is marginal and not worth spend the extra $20 for.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113280
AMD'a APU also benefits from fast RAM. Getting DDR3-1866 would be the best option as mentioned by
8350rocks. There is even faster RAM, but they cost more and the benefits are marginal at best. Besides, you are going from about $75 to $130 if you decide to choose the A10-5800k vs. the A8-3670k. I recommend 8GB of RAM since the APU uses some of the system RAM for graphics; such as the following for $70.
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231550
The total cost excluding the motherboard is $200.
----------------------------------------------------------------
As an alternative, you can opt for a dual core Intel CPU since the games you listed do not make use of more than 2 cores. Intel CPUs are clock for clock more powerful than AMD CPUs and APUs. For example, click the link below which compares a dual core Penitum G850 to the A10-5800k. The game benchmarks is based on whatever discrete graphics card is used for both test rigs (same GPU for both). As can be seen, the dual core Pentium G850 is better than A10-5800k when a discrete graphics card is used.
http://www.anandtech.com/bench/Product/404?vs=675
However, I recommend the newer Pentium G2120 which is not only 200MHz faster, but in general, the average performance increase over the prior generation (assuming the same clockspeed) is 6%. The G2120 (3.1GHz) costs $80; alternatively, you can buy the G2020 (2.9GHz) for $65.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116777
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116886
Because you will need to buy a discrete graphics card for the Pentium G2xxx, you do not need to buy 8GB of RAM. 4GB will suffice an Intel CPUs do not really benefit from very fast RAM so the overall cost is less. You can get the following 4GB for $40.
http://www.newegg.com/Product/Product.aspx?Item=20-231-276&SortField=0&SummaryType=0&Pagesize=100&PurchaseMark=&SelectedRating=-1&VideoOnlyMark=False&VendorMark=&IsFeedbackTab=true&Keywords=%28keywords%29&Page=1#scrollFullInfo
To keep costs down just go with the Pentium G2020 and the 4GB of RAM which works out to $105. Compared to cost the A10-5800k + 8GB of RAM, you have $95 left to spend on the graphics card to come to the same $200 total cost.
The best graphics card you can buy is the Radeon HD 7750 w/ DDR5 RAM for $85 after a $10 rebate.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102969
That brings the total cost to $190; Radeon HD 7750, Pentium G2020 and 4GB of RAM.
Alternatively, you can buy the Radeon HD 7750 w/ 2GB of slow DDR3 RAM for $80. But you don't really get any benefit from 2GB of RAM unless you play above 1920x1200 resolution. The Radoen HD 7750 is not powerful enough for that. DDR3 vs. DDR5 probably means a 6% - 10% decrease in performance; not worth it to save $5.
If you really want to save money on a graphics card then consider the following Radeon HD 6670 w/ DDR3 RAM for $50 after rebate.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102988
Going with the Pentium G2020, 4GB of RAM and the Radeon HD 6670 works out to $155.
===================================
So... how do the different graphic cards compare?
The integrated Radeon HD 7660D in the AMD A10-5800k is a little more powerful than a desktop Radeon HD 5570. It's considered more of a multimedia card rather than a gaming card.
The Radeon HD 6670 w/ DDR3 RAM is about 6% - 10% slower than if it came with DDR5 RAM, but it is still faster than the Radeon HD 5670 by a decent amount. This card is considered the absolute minimum for a budget gaming rig.
The Radeon HD 7750 is more or less equal to the Radeon HD 5770 / HD 6770 (same card; different name), and is significantly more powerful than the integrated Radeon HD 7660D graphics core.
Whether you go with the $155 configuration or the $190 configuration of the Pentium G2020 + 4GB of RAM + whatever GPU, the gaming performance will be much higher than the A10-5800k and it's integrated graphics core.
Even if you were to play Battlefield 3 multiplayer which is known to benefit from a quad core CPU, the difference in graphic card performance means that the Pentium G2020 will perform better in BF3.