as mentioned, you'd be further ahead buying a .22-250 rifle. Lots and lots of fine .22-250s out there, from many makers. Then if you want, get a custom barrel for it with the twist rate to suit you.
The .22-250 was designed as a varmint round. Its decades older than the .223 Rem and beats it in velocity by 500+ FPS. Its optimized for shooting 50-55gr varmint bullets and does a really good job at it.
If you want to shoot somthing heavier, for larger game, where legal, or for long range target shooting like is popular with the AR crowd these days, the factory twist rate barrels (1-14" usually some are 1-12") isn't going to work.
You can get a custom .22-250 barrel in the faster twist rates, and that would work just like a fast twist .223 with much higher velocity.
Converting a .223 rifle into a .22-250 simply isn't cost effective. As mentioned, besides the .22-250 barrel, you'll need the correct size bolt, properly fitted by someone who knows what they're doing, AND action work on the magazine box and feed rails, again not something for the home gunsmith unless you just want to ruin a good gun....
you'll very likely spend considerably more on parts and action work to do the conversion than you will spend buying a good complete .22-250 rifle.
Especially at today's prices....
Get a Rem, Win, Ruger or other good .22-250 rifle and be happy. Don't waste your money on a very difficult and expensive conversion of a .223.
Assuming you can even find a smith willing to do the work...
One more point, if you are interested in a fast twist .22-250, ammo will be a handloading proposition, All the factory ammo is loaded with the 40-55gr varmint bullets.
Get another rifle, get a used one if you can (save a few $) and if the barrel is toast, rebarrel that rifle and its new all over again!