As Aguila said, be sure to read the handgunlaw.us page. Alaska is famous for constitutional carry, but they do have some restrictions you might not think of, eg getting a homeowner's express permission before bringing a gun into his house. Also, IIRC, brass knuckles and nun-chucks are verbotten. Be aware that there are several locations in Anchorage which are posted "No Guns". Shopping malls, performing arts theatres, etc. Consider using dry lube on any gun that's going to be cold-soaked; even gun oil gets thick at some point. Assuming you conceal your carry piece, it's probably not an issue; now you just have to figure out how to draw from under a heavy coat. I saw a gun jacket once that only zipped from just above the waist upwards, leaving the bottom loose enough to pull aside for easy access. Can't remember the brand, but looked like a neat idea.
If you're in the military, I expect you have to wear what they give you? Otherwise, plan on spending $400 for clothes.
Indera Military Fleece long johns from Walmart.
Carhartt hoodie. (yes I know it's $70.00
but it's worth every penny.)
Refrigiwear hooded overall from Alaska Industrial Hardware.
Refrigiwear gloves and mittens, also from AIH.
Bunny boots from G. I. Joe's Surplus at 5011 Arctic Blvd. $50 cheaper there than the big Army/navy store.
There are pros and cons between an overall and an insulated pants/jacket combo. Take your pick. But I would highly recommend that the hood on the jacket be permanently attached; the snap-on hoods allow plenty of room for the wind to blow up the back of your neck. That's why I went with Refrigiwear over Carhartt.
Don't let the locals convince you that you only need bear spray or bells. Familiarity breeds contempt.
Carry a .500 S&W for the bears, and a 12 guage for the mosquitoes.