This is probably the cheapest system I'd build today for games.
Rosewill Blackbone Black Steel / Plastic ATX Mid Tower Computer Case
Model #:Blackbone
Item #:N82E16811147023
$49.99 -$10.00 Instant $39.99
I've generally liked any Rosewill case I've used. I've built in this one. It comes with more fans than near competitors, and they are quiet.
ASRock B75 PRO3-M LGA 1155 Intel B75 HDMI SATA 6Gb/s USB 3.0 Micro ATX Intel Motherboard
Model #:B75 PRO3-M
Item #:N82E16813157329
$79.99 -$10.00 Instant $69.99
I've also had good results from ASRock, especially on a budget.
MSI R6770-MD1GD5 Radeon HD 6770 1GB 128-bit GDDR5 PCI Express 2.1 x16 HDCP Ready Video Card
Model #:R6770-MD1GD5
Item #:N82E16814127599
$79.99 $79.99
This is a great deal for this card right now. It is notably better than the HD6670 you were considering, and actually may be cheaper than some.
Antec NEO ECO 400C 400W Continuous Power ATX12V 2.3 / EPS12V 80 PLUS Certified Active PFC Power Supply
Model #:NEO ECO 400C
Item #:N82E16817371029
$59.99 -$15.00 Instant $44.99
Unlike the Corsair V2 Builder, this is a Seasonic-built PSU. Be aware that it does not include a power cord, so you'll need to re-use one or buy one.
Intel Pentium G850 Sandy Bridge 2.9GHz LGA 1155 65W Dual-Core Desktop Processor Intel HD Graphics 2000 BX80623G850
Model #:BX80623G850
Item #:N82E16819116397
$69.99 $69.99
There's wiggle room in the budget to move up to an i3 here (and what I would do), but like I said this is the cheapest system I'd consider building for games. Even this Pentium performs as well or better than most stock AMD CPUs in games.
G.SKILL NS 4GB (2 x 2GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10600) Desktop Memory Model F3-10600CL9D-4GBNS
Model #:F3-10600CL9D-4GBNS
Item #:N82E16820231394
$19.99 $19.99
I've had good luck with G.Skill. 4GB is enough on a budget, but I've seen some pretty good deals on 8GB.
Western Digital WD Blue WD5000AAKX 500GB 7200 RPM SATA 6.0Gb/s 3.5" Internal Hard Drive -Bare Drive
Model #:WD5000AAKX
Item #:N82E16822136769
$79.99 -$10.00 Instant $69.99
If you prefer a five year warranty, get a WD Black series drive. They also perform a little better, although if it's performance you want, I'd suggest getting a SSD (minimum 120GB to 128GB; you WILL use the space on it).
LG DVD Burner Black SATA Model GH24NS90 - OEM
Model #:GH24NS90
Item #:N82E16827136247
$17.99 $17.99
Meh. One optical drive is much like another.
Subtotal: $412.92
Like I said, there's wiggle room. A better CPU would likely be my first upgrade, especially for MMOs or multiplayer games.