Here is a link to info on the legality suppressors and NFA weapons:
* Liberty Suppressors NFA FAQ Page *
There is a $200 Federal Tax Stamp required for a suppressor. If you can pass a background check for buying a firearm, you should have no problems passing the background check for a suppressor. A suppressor would have to be transferred though a Class III Firearms Dealer.
Currently, here in MS we can hunt with suppressors. Every state has different laws.