Hi! Since you want to play skyrim and BF3, I will be using those as guidelines on "building" your computer. You can view what I'm using here:
http://www.tomshardware.com/reviews/battlefield-3-graphics-performance,3063-10.html
and here:
http://www.tomshardware.com/reviews/skyrim-performance-benchmark,3074-6.html
Breaking it down:
Since you have a budget of ~$650, you can't have all the bells and whistles in your system, so by looking at the bench marking results, Skyrim can be played at a decent 40FPS with a 1920 x 1080 monitor on ultra using a Radeon 6850 and with the 6850, battlefield would sustain a feasible, i would say, 45FPS with FXAA on. For the CPU, we'll go with the AMD Phenom II x4 955BE, because of 2 factors. 1: it is overclockable and 2: it is cheaper than Intel's i3
and keep in mind that the 5850 is only a tad bit better than the 6850:
http://www.tomshardware.com/reviews/fastest-graphics-card-radeon-geforce,3067-7.html
CPU: AMD Phenom II X4 965 Black Edition $120
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=5752143&CatId=4431
Graphics: HIS H685FN1GD Radeon HD 6850 $145
http://www.newegg.ca/Product/Product.aspx?Item=N82E16814161384
MoBo: Foxconn M61PMP-K AM3 $50
http://www.newegg.ca/Product/Product.aspx?Item=N82E16813186189
PSU: Cooler Master eXtreme Power Plus 500-Watt Power Supply $40
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=4465483&CatId=1079
Case: Thermaltake VM30001W2Z V4 Black Edition $35
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=6447916&CatId=1509
HDD: Western Digital WD5000AAKX Caviar Blue Hard Drive $110
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=7281716&CatId=139
RAM: Crucial CT2KIT25664BA1339 4GB $32
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=4779409&CatId=3361
Heatsink+Fan: STOCK $0
DVD-ROM: Sony Optiarc AD7280S-OB 24x DVDRW Drive $20
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=1387814&CatId=89
Maybe:
OS: Microsoft Windows 7 Home Premium $110
http://www.newegg.ca/Product/Product.aspx?Item=N82E16832116986
If not:
HIS H687FN1GD Radeon HD 6870 instead of the 6850 $170
http://www.newegg.ca/Product/Product.aspx?Item=N82E16814161389
Total:
w/ OS: $662 + shipping
w/ 6870, w/o OS: $587 +shipping
Explanation:
The AMD phenom II x4 is a great CPU, has 4 physical cores and is one of the best processors you can get for the money. The motherboard is cheap, yet has decent reviews. The PSU is from CM, which is a reliable company, case is cheap. A 500GB hard drive for that price is a fairly good deal since the flood that brought the prices up and the RAM is fairly priced too. Since you wanted an Intel CPU, you could have gone with the i3 2100 for a bit more money and a intel mobo. something such as:
CPU:
http://www.tigerdirect.ca/applications/SearchTools/item-details.asp?EdpNo=7295297&CatId=6983
$130
and
Mobo:
http://www.newegg.ca/Product/Product.aspx?Item=N82E16813131787
$70
The difference? The intel CPU has 2 physical cores, but can have 4 threads (just like the Phenom II), and it is slightly better performance wise. So if you don't need windows 7, you can go and buy the Intel CPU, the Asus MoBo, and the 6870 for about the same price as if buying the AMD set with Win 7