You don't need a license or a permit, per se - what you will have to have (as long as suppressors are legal to own under Alaska state law) is a "tax stamp" which is done by paying $200 to the government, which will allow you to legally posess a NFA item such as a suppressor.
First, find a Class III dealer in your state. Most of them will be able to walk you through the rest of the process. There are also a few ways to go about it (forming a Trust, etc), with different people advocating different approaches. All work, which one works best for you, however, is something that you're going to have to figure out for yourself.
But in short - you do have some hoops to jump through, but in the long run, the process is not that difficult. I'm considering getting into the NFA game myself as soon as I Can afford the tax stamps for a SBR and suppressor to go along with it.
Good luck!