What resolution 1920x1080 ?
How much anti-aliasing ? What type of anti-aliasing ? MSAA is nicer, FXAA is uglier, but faster. 4xMSAA would be good, I think. Do you also want transparency AA ? You can get 16xAF easily, it doesn't cost much. But what about Ambient Occlusion (SSAO) ? It's very nice to have it.
What framerates do you want ? 60 fps at all time ? Or do you not mind to dip below at some spots (like at the stairs in White Run).
What country do you live ? The world is big.
What is 1G ? 1K ? 1K what ? Euros ? Dollars ? What kind of dollars ? US dollars ?
You gotta be specific, or you will get useless answers.
Before you answer, the only thing I can say is, building your own desktop is the cheapest. And if you've never done it, it's a lot easier than you might think. Many US people buy their PC components at newegg.com. Good prices, and a huge variety of stuff you can buy. (You can basically buy any component there). Have a look. For a small fee, they might even assemble the PC before they ship it to you.
One more suggestion.
This would be my gaming machine.
1) CPU - Intel i5-3570k. Released end of April. $225.
2) Any Z77 motherboard. $100
3) 8GB of RAM, ~$40
4) Proper PSU, lots of choices. $50
5) Case. $50-$100.
6) HDD. SSD. DVD-burner. Varies on sizes. Roughly $50-$100.
7) The graphics card. Most important part. $200 will get you a nice gtx560ti. $500 will get you top, gtx680.
To summarize. For $750 you'll have a PC with a gtx560ti (or equivalent AMD card). It'll be a very fast PC. And excellent at playing games. I think you can play Skyrim on Ultra at 1080p with 4xAA.
For $1050 you'll have a machine with a gtx680. Basically the fastest gaming machine money can buy today. It'll play Skyrim with all eyecandy you throw at it, texture packs, 8xMSAA+Transparancy AA + high SSAO, etc. Definitely worth the money, I bought a gtx680 last week (500 euros) and I love it. In 3 weeks, the new Intel CPUs will be released, and I'll buy a 3570k myself. My machine will be awesome.