I have built exactly one rifle in my life, and it is very similar to what you desire -- budget oriented, light caliber, and really sweet shooting at the ranges you mention.
Action was a 700 ADL, seventies vintage. (222 Rem) A local smith turned the old barrel out, and I had to force twenty bucks on him for the service.
I ordered a barrel chambered in 20-222 from PacNor, Remage type, and IT IS BEAUTIFUL. Price came in at $550, which didn't seem cheap at the time. I know different now. The precision fit of those threads into my unaltered action was a beautiful thing.
I had lucked out and found 222 Go/No-go gauges on eBay for twenty sumpthin. -- definitely in the budget zone.
I ordered a stock from Numrich without much expectations because it was dirt cheap. (like 53 bucks shipped) It was an old 40X rimfire stock. Not a full-on bench rest stock, but as close as I cared to go. The thing was freakin great. It fit my action perfectly, and took minimal work to open up the barrel channel to fit the barreled action. I was able to get the proper bottom metal for it, for under forty bucks, which made the project easier, add a new Pachmeyer butt pad, for $15, and I had my stock for a little north of a hunnert bucks.
I'll cut to the chase, and say that this thing shoots great, easily sub .5 MOA, and I have many groups in the .2's and .3's. Haven't done anything with the trigger yet. This one breaks at 3 lbs. and isn't bad but that will probably be a future expense.
Everything came together perfectly with this rifle, and even after the addition of a Vortex Viper, I'm in at less than a grand. jd