I'll post my computer. I was in the exact same price range with the exact same goal and I achieved it with great success.
XFX Radeon HD 7870
http://www.newegg.ca/Product/Product.aspx?Item=N82E16814150605
I went with AMD because they are cheaper and really it gives you the same power. The next tier video cards were another 80 bucks or so more then this so I just settled with this, this is overkill right now for games, and 2 free games with purchase was quite a nice bonus =)
Seasonic 620W Modular power supply
http://www.newegg.ca/Product/Product.aspx?Item=N82E16817151095
From what I read, the PSU is the part that is most likely going to blow up in your face and get you pissed right off. Seasonic and Corsair were names that were well respected, so I was looking for one of those. I read that Seasonic actually makes Cosair's cheaper models for them, and I found this modular one which I thought would be good for me because I was so newb at building PCs
Corsair Vengence 8 GB Ram
http://www.newegg.ca/Product/Product.aspx?Item=N82E16820233309
Everything I read pointed me toward Corsair RAM, its what there company is built on.
MSI Z77A-G45 LGA
http://www.newegg.ca/Product/Product.aspx?Item=N82E16813130645
Honestly with the motherboard I had no idea what I was doing, I kind of picked everything at random but this had decent enough reviews and the brand is respected so I went with it, everything turned out perfect.
Rosewill Challenger Mid-size comptuer case
http://www.newegg.ca/Product/Product.aspx?Item=N82E16811147153
I wanted to get a prettier looking one but if I did that I would end up paying 20-30+ shipping rates which I was not okay with. I went with this one, keeps my computer at 30'C idle temp and never goes above 50 even in the most load-intensive games.
Intel I5-3570K Ivy Bridge
http://www.newegg.ca/Product/Product.aspx?Item=N82E16819116504
Intel is the only processor you should look at, nuff said.
I did hours and hours of research on what components I could get that were the most bang for my buck, and more importantly what was the most reliable (I never knew how sketchy it is to build a PC because parts will blow up on you or arrive dead) ,this is the result. Keep in mind these are Canadian prices on an American-based Canadian site, so if you live in America you could probably get the same thing for a couple hundred bucks cheaper. All in all, JUST the computer cost me around 800-900 at the time if I remember correctly. With moniter/keyboard/mouse/headset I think the total price tag on my rig is around 1100-1200.
With this I can completely obliterate Skyrim, with extremely heavy modding (30-50 mods at once). I can play every single game I've ever tried on Ultra (except for Crysis 3 but I'm thinking it had something to do with the type of video card and not lack of power and even then it was very very close to being able to do it). It runs so smooth, so fast, and so quiet. It was my first build and it couldn't have gone better really. I had one hiccup with my video card but was easily fixed.