Fitting a barrel to a centerfire rifle starts with getting a rifled blank. That blank may or may not be profiled and cut to the length one desires. If not, it can be turned down on a lathe to the profile one needs. The best way is to put a live center in both ends so the outside will be turned concentric with the bore. Turning the barrel's diameter to a smaller one will do the following to its bore and groove diameters; if the rifled blank is:
Crowing the barrel's front end may be next. Holding the barrel's breech end in the lathe's headstock, its muzzle end is held in a steady rest. Best crowning is attained when the barrel turns on its bore axis. Precision live centers are oft times used in the muzzle so that part turns well centered on the lathe's tailstock to turn down the muzzle's outside diameter. Once that's done, that part is put in the steady rest and the bore then turns on its center. The muzzle is faced at some angle to the bore with the cutting tool starting inside the bore and moved out with the lathe's crossfeed. 11 degree facing tends to produce the best accuracy, but as long as it's straight, that's fine. Some folks put a rounded muzzle face on the barrel so they look like factory barrels. After facing the muzzle, the crown on the rifling is done. A brass ball about twice the bore diameter has lapping compound on it and it's pressed into the bore just enough to deburr the rifling and make it smooth and uniform. Some folks use different tools, but they all do the job.
Once the barrel is profiled, shanking it is usually the next step. The muzzle end can be held by a lathe's headstock, but the breech end is usually held in a three-point steady rest a few inches in front of where the barrel tenon, or shank will be cut and threaded. If the barrel's back end diameter is concentric with the bore axis, then the bore will spin well centered. If not, then a live center should be put in the bore's breech end and the outside diameter turned to get it trued up, then put that part in the steady rest.
The diameter, thread count, and length of the barrel shank needs to be determined. That part is then turned down and threaded. If an extractor cut needs to be made on one side of the barrel's breech face, that can be marked by putting the barrel in the receiver, then marking where the extractor will go. Extractor cuts are made on a milling machine or a milling attachment put on the lathe's crossfeed.
Chambering is the next thing. Again, the breech end's in a steady rest. Some folks with bigger lathes have the muzzle end sticking out of the back of the headstock and held there by a three or four jaw chuck; the chamber end is held by the front chuck. Regardless of the setup, it is important that the barrel blank turn well centered on its bore at the chamber end.
A roughing reamer is used to remove most of the chamber parts metal. The actual chambering reamer is then used to ream out the chamber to fit the cartridge. As the distance from the closed bolt's face to the headspacing part of the chamber is critical, a good machinist will determine how far the chambering reamer needs to be put in the bore.
Chambering reamers are made in two types; solid or floating pilot. A pilot is what centers the reamer on the bore's center. Solid pilot reamers have a round part at the chambering reamers front end. That pilot is a few thousandths of an inch smaller than bore diameter. Floating pilots are screwed into the reamer's front end; they come in different diameters. If the pilot's diameter is only a few ten-thousandths of an inch smaller than bore diameter, the chamber's throat will be very well centered in the bore; very important for best accuracy. Floating pilots turn with the barrel and don't marr the bore in front of the throat and leade. Solid pilots sometimes do marr the bore and oft times don't center the throat and leade very well.
After the barrel is shanked and chambered, it's screwed into the receiver and torqued up. One doesn't need a lot of torque on the barrel. A 2-foot long action wrench is enough; some folks use shorter ones. Headspace is checked and if it's too short, the barrel can be removed and the chamber reamed a tad deeper. Sometimes, the chamber reamer can be put through the receiver on an extension, then the chamber deepened the correct amount.
I didn't mention standard machine shop practices regarding how fast the barrel turns, lubricating the reamers, how fast to push them into the bore and finally keeping the chamber clean and the chips removed frequently. That's something one needs to do correctly.
The order these things are done isn't fixed. You can chamber the barrel before shanking it. You can turn down its outside diameter after it's shanked and chambered.