There are plenty of online articles explaining the process. This is one, but doing a Google search will find others.
Basically, you find the suppressor you want, either at a local dealer, or from one of the many online shops like Silencer Shop, Capitol Armory, or Hansohn Brothers, and buy one, which starts the process. You fill out the paperwork and send it in, along with photos, fingerprint cards, and a check or MO for $200 to the ATF, and be prepared to wait for up to a year before you get approval to take your new toy home. The dealer should be able to help with the paperwork and guide you through the process.
Youíll have to decide whether you want to file as an individual or a trust, and if itís the latter, you need to get the trust set up beforehand. Again, plenty of search information about whether individual or trust is better.
As a note specific to the Osprey, youíll need to buy the properly threaded piston separately because the Osprey doesnít come with a piston.
Personally, my first purchase was from a LGS, while subsequent ones were through Silencer Shop.