The first thing you need to do is list (itemize) every little thing you're gonna want in the new rifle - exact trigger, barrel (maker/length/contour/material/features like fluting, etc), stock, scope & mounts, and who's going to do the work. You're not gonna believe what you're gonna end up with, and what it'll cost, new !
If you're "on a budget", you can save huge amounts if you can do much for yourself.
If you decide to go with a used rifle as a basis, find one with as many features on your list as you can - for instance, Remington 700's are more expensive, but they come with one of the better factory triggers available, as does the Winchester M70 and the newer Savages with the Accu-Trigger. It's arguably easier to find & bed (read: cheaper) a custom stock for them, too.
An example would be to locate an early M700 Varmint (heavy Barrel) in one of the smaller .22 centerfire rounds, and simply have it rechambered for a .22 Super Duper Whatever.
If money is an issue, then you'll be several hundred dollars better off just purchasing a new factory super whatever, like a .204 Ruger.