Okay - I own a .22-250. Nice flat shooting, and powerful. I also own a .22PPC and a .223...
If you handload, and are having the rifle made, get either .22PPC, .22 BR Remington, or .222 - You can get BETTER brass - You'll spend a little more, but accuracy costs. The .222 holds the world's record for smallest 5-shot group at 100 yards (some feller named Gale's brother...). I'd probably go with .222 or .223 Improved (not the regular chamber - less brass stretching with the improved version).
For under 300 yards, I'd go with the deuce (.222), or the .22PPC... For 300-400, I'd go with the .22 BR, or maybe the .22-250... Out to 500, a .22-250 Ackley... Over 500, go with a big six...