With that budget you're looking at a Savage 10, Tikka T3, or Remington 700 in .308 Win. Check out all those rifles and choose the one that feels best to you, I personally like the Tikkas as they are factory guaranteed to shoot 1 MOA @ 100 yards.
Then you will need a scope, look into the Bushnell (elite?) 10x or the SWFA 10x (or 16x). There are lots of good scopes out there, but these two have excellent tracking and good warranties.
Keep in mind, you wont be able to get good @ 1000 yards or even 800 if you don't practice at closer ranges first. A lot of people say "I want to shoot 1000 yards" without having shot 600 or even 400. It takes a lot of practice and knowledge to get out to those distances. In fact, if you really want to maximize your setup, you WILL have to reload your own ammo (or spend a fortune on custom match loads).
I hope this gets you started, I'm sure others will have other opinions.
Good luck!
- TNG
Edit: If you're going to reload, look into getting a .270. The ballistics of that caliber are very good. Ask around about other calibers too, there are PLENTY that offer more than the .308. For example, if you are going to be hunting with it, you might find a rifle in .30-06 more useful as that round is capable of launching anything 220gr boulders for large game to light 125gr loads for deer. The .308 is just what our government and NATO has chosen as their "go to" extended-range round.