You purchase the suppressor. If you purchase it locally, no problem, if you purchase it from an on-line dealer, that dealer will transfer the suppressor to your local dealer vis a Form 3.
You complete a Form 4, either electronically or in paper form. BATF just recently reopened the E-File for Form 4's.
Submit that Form 4 along with the $200 tax, a passport type photo, fingerprints, and then the wait begins.
I recommend you seek out a dealer that has a Silencer Shop kiosk for your first time.
You will also need to decide if you want to acquire the suppressor as an INDIVIDUAL or on a TRUST. The folks who have the Silencer Shop Kiosk can help you with Silencer Shop's "ONE SHOT TRUST" and explain to you the advantages of the One Shot Trust.