As already mentioned, if you're driving through Canada you can't bring handguns. You can ship them to yourself via FFL holders but you'll have to pay transfer fees and go through a background check on the receiving end. I've lived and hunted in Alaska for 12 years and I've never once needed a handgun. I occasionally carry my 10mm Glock 20 but that's mostly because it makes my wife feel warm and fuzzy in bear country. Any decent big game caliber works fine up here. Shoot what you're comfortable with. A 30-06 will kill anything in AK.
Get a copy of the hunting regs to get specifics. Alaska has 26 game management units and most have numerous subunits, that all have different regulations. Black bear is open in a lot of the state year round. Most caribou seasons start in August, and most moose seasons start in September. There's a ton of variables so specifics about your plans can give you better answers.