The federal process is to obtain the necessary form, submit it along with a photo, two fingerprint cards, and your $200, to the ATF. ATF will process, FBI will run your prints, background check will follow.
Your local police chief, sheriff, or district attorney has to sign the form before submission.
After anywhere from two to six months, if you're clean & the paperwork's in order, it'll come back to you with a tax stamp attached, and you're in business. You then go pick up or build your gun.
The paperwork is specific to the gun you're looking at buying or building, it has to have the info on that gun listed on the form.
You find a gun you want to buy or build/convert, then do the above paper process.
You do the same process on each & every Class III gun you buy.
There is another way around most of this, you can also create a corporation which applies for the tax stamp, bypassing background check & fingerprint cards. But, most go the regular route unless your local LE people won't sign off on you.
There are some state restrictions, not legal to possess by state law in one or two.
Denis