There is no such thing as a "Class III permit".
You need to be an adult with a clean record, and live in a place where they are legal.
The cost is $200 for the Tax Stamp and the cost of the suppressor itself.
Usually the hardest part is getting the Chief LEO in your town/city to sign off on it. Even if s/he will not, you can still get around that by incorporating, but that's more time and money.
Assuming you hit no snags, the timetable is about 3-6 months.
Read this for good general info:
http://www.ar15.com/forums/topic.html?b=6&f=17&t=199928