For a pellet barrel, a standard reamer at 0.1 to 0.15mm over the nominal caliber works well. So for example in .22 cal (5.5mm), a 5.6mm is a good choice for reaming the leade. It will just wipe out the rifling. Then break the sharp leading edge of the rifling with a rubberizied polishing bit and/or wet/dry sandpaper so the pellet head can swage gently in the rifling rather than getting shaved.
For a slug barrel, a slight taper of about 1° is preferred. Custom barrel reamers are pricey but a tapered pin reamer (1.2° included angle) is a good and inexpensive way to do it. Figure out where on the taper is the maximum diameter you need and grind away the portion behind it. For your .223, a #3 pin reamer would be good. It tapers out to a max of 0.229" so you wouldn't have to take off much. For a one-off barrel, run it in somewhat short of what you think you'll need and try seating a slug. Sneak up on the depth you need, such that your bolt probe seats it just into the taper.