I think the only thing that I would keep from that build is the optical drive. Are you going to be using this computer for anything else?
If you are not going to be overclocking then you don't have to get an aftermarket heatsink. But if you are going to OC, then get either the Cooler Master Hyper 212 Plus or Scythe Mugen 2 (Rev.B if you stay with LGA 1156).
You should only really go i3 if you want to use the on-chip IGP. Which is not that useful for gaming. Online gaming it would be fine with.
If you can't find the 500GB Samsung Spinpoint F3, then get the Seagate Barracuda 7200.12 as it is a close second to the Samsung in performance.
The GTS 250 is not a very good card, I would try and get the 5770 at least. The 5750 is very similiar in performance to the 250 and uses a lot less power. However the card choice is determined by the resolution of your monitor as well as the games you want to play.
If you do change to an AMD system, then you can get a less expensive motherboard than the one bnot mentioned: Asus M4A79XTD Evo Or Gigabyte GA-790XT-USB3. These boards don't support RAM above 1800Mhz, so you would have to get slower RAM in this case if you didn't want to waste money. They both support crossfire tho. And the Gigabyte one has USB3, which isn't going to be useful for awhile.
Instead of the Athlon II X4 620, the Athlon II X3 435/440 are very solid in games and a bit cheaper.
I wouldn't go for that PSU as it is not from a company that I know to produce quality power supplies. A 500W from a good brand such as Antec, Corsair, Seasonic, OCZ, FSP, Silverstone, Zalman, XFX would be fine for one video card.
Do you need a case?
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