I asked the exact same question of my way up there supervisors supervisor at work. He claims to have been an Army sniper. He response left me grinning, but educated at the same time.
He said if you want to do it on the cheap, the Savage 110 FP would do the job nicely. You can probably get the gun for about 500 dollars and then you would have to get a good scope. Anything 12X or better at max resolution would work. He said because the ballistic tables are so easily gotten, he would recommend .308, but out to 1k yards, the .338 lapua held the records for tightest groups. He told me to practice at 100y, and when I could consistently get .5 moa groups at will, then I would have the craft to shoot the longer distances. He recommended the 110 FP because it came with the bull barrel.
With that said, I have started collecting .308 brass and ammo in prep for my next purchase. I grinned, because having shot 400y before (and I very rarely missed my target at 300 or 400 using my saw or m16) I know there is more to shooting 400y then there is at 100y. I have watched the wind take my tracers from left of the target to right (I thought I had adjusted enough). Next two round burst got it, but it still left me confused. Wind and at longer distances (say 700y +) curvature of the earth become real considerations. Wind is a bust at 100y with that type velocities. Even the Army is supposed to shut down ranges if there is sustained 10 + mph winds (note I said supposed) because it can make your rounds completely miss at 200y (which is just moving it 6 inches from center to left or right of the target).
The scout sniper I used to work with (I am a contractor for DoD, most of us are former military, myself included) also recommended .308. He has volunteered to teach me to shoot out to 1k, if I can ever get a rifle and range that will do. He personally uses an accuracy international, but that was 1k plus the 1.2k for his mark IV scope.
