Suppressors are fun to shoot and can be owned by purchasing from a class 3 dealer in your area.
You decide what you want to own and purchase it. Your dealer will usually fill out the transfer form for you. You will need 2 sets of FBI fingerprint cards and a set of passport photos. You sign a form certifying you are a citizen of the US.
The forms are sent in to the local sheriff or police chief to certify there is no state law prohibiting your having a suppressor.
The forms, once signed will be returned to you. The dealer will review them for completeness and mail them off to the NFA Branch of the BATF along with your check for the one time $ 200 federal transfer tax.
They run your prints and background check and return the approved form in about 90 days with a tax stamp attached, to your dealer. He calls you and you come in and complete a form 4473 and pick up your suppressor.
Keep the original form somewhere safe and carry a copy with you when you have the suppressor on an outing.
If you currently have passed a NICS background check, you will pass the BATF background check.
That's the simple answer. You may want to look at sites like
YHM, Yankee Hill Machine Co., Inc. ï¿½ Makers of High Quality Firearms, Accessories & Sound Suppressors | Sound Suppression - The Future of Sound Suppression for samples of quality suppressors.