Welcome to the Forum -
1,000 Yard shooting is one of the hardest things I have ever done. And if you are starting out, I would suggest setting your "sights" a little lower. One does not simply JUMP into shooting 1,000 yards... LOL
I assume you want a bolt gun? Correct?
And you want the weapon and the scope to come in at $650??
Don't want much, do you??
The first thing you need to do is decide on a caliber. From there, we can work on the weapon and the glass.
The .308, mentioned above, is a good choice, but the maximum effective range for that weapon is really about 650 yards. And at that, it's quite a moon shot because the bullet drops like a brick.
And before some yay-who comes along ranting and raving about how the Marines shoot 1,000 yards with their M24 in .308 all the time - you're right, but that is because they HAVE TOO. They are not retooling the entire Corp, or the Army, with new weapons at the moment.
There are NO .308's going to the sand box at the moment. Everyone is taking, at a minimum, a .300 Win or a .338 Lapua
The FBI's HRT Team went to the 7mm Remington Magnum about 6 or 8 years ago and there is a thread some moron is building on that cartridge in the Projects Section
So, we need to decide on a cartridge first. What do you want the round to do? Realistically? What are you expecting out of the weapon? What kind of ranges do you have around your home?
There are only a couple of places that we can even shoot further than 300 yards around here, so building a 1,000 yard gun is something that you need to plan on being able to use...
So, provide some insight and we can go from there.
What books have you read? I might be able to suggest some great reading material on the subject, I have an extensive library...