Building a 1,000 Yard Rifle on a Budget is not the easiest task in the world, but it definitely is possible! A true thousand yard gun requires three things: Accuracy, Repeatability, and a decent Rifle Scope.
First off, when buying the rifle you plan to start with, you can save some money by picking up a used gun at a gun show or from a dealer that stocks used rifles. You'll want to know how to inspect the rifle before you make your purchase though, because even tiny imperfections can make life difficult when you're building a long range rifle. You'll also want to give some consideration to the rifle's caliber, because not all calibers can reliably reach the 1,000 yard mark. I generally suggest a Remington 700 in .308 Winchester for new shooters, as both the rifle and caliber have already been proven to work at the ranges you'll be shooting. In any case, a high quality used Remington 700 should cost around $500. If you don't want to spend that much, a Weatherby Vanguard or Mossberg ATR is a fine second choice with a price tag of under $500.
Before you can test your new rifle, you'll need a Rifle Scope to put on it. While many modern bolt action rifles come equipped with Iron Sights, which work in a pinch, they are virtually useless for anything over a few hundred yards. When it comes to the scope's features, you'll want something that has a decent zoom level (10x or more depending on your comfort level), as well as a reticle you are comfortable using. My reticle preference is the standard mil-dot scope, but you can feel free to choose whatever you'd like. If you will be shooting at predominantly unknown ranges (places where you won't know your distance-to-target 100%, I suggest something capable of range finding such as the mil-dot (preferrably mil-dot in First Focal Plane, but that is also up to you. A Mil-Dot scope in the First Focal Plane can be had for as little as $230.
Once you have your rifle and a scope, you'll need to put the two together. If your scope came with Scope Rings, Great! If not, you'll need some. I suggest a high quality set of steel scope rings, but you can use an equally high quality set of aluminum rings if you prefer. Just make sure you don't skimp on quality or price too much here, as even the best scope is fairly useless with a $2.00 set of bargain bin rings. Scope rings should cost no more than about $70 per pair.
So far, that's $500 (or less) spent on a rifle, approximately $230 spent on a scope, and $70 spent on rings. The total? Just $800! You can easily spend less on the scope if you aren't too picky about features ($70 to $100 can net you an acceptable "starter" scope), but I hope I have shown you that Building a 1,000 Yard Rifle on a Budget is entirely possible!