This topic is a very important one if you are DIY hunting Alaska and want to get your valuable meat home... it should be budgeted in to your trip, especially if you're hunting moose or multiple animals. I've done six successful DIY trips to Alaska and believe that the way I did it was the cheapest at the time... airline rules an fees change all the time. My experience is with Alaska Airlines and Alaska Airlines Cargo.
With that being said... the cheapest way is to take it home with you on your commercial flight as extra baggage., just like Doug mentioned. Follow the airline packaging, weight and size limits and you shouldn't have a problem. Tons of fish and meat are flown out of Alaska so they're used to it if you follow the rules. A boned out Sitka blacktail will fit nicely in to a 50# fish box. A caribou will take up about two 100# ice chests. It doesn't take much volume of meat to 100#. I suggest weighing your bags in at least 2# under the max weight limit and arrive at the airport way before your departure time. On more than one occasion I was pulling meat out of a box in the Anchorage airport because of their corrupt baggage scales. Both times I weighed the ice chests at AA Cargo and they were under.... go figure. Nothing wrong with bribing the counter lady with some caribou or moose meat.... it worked both times.
Getting large antlers and a moose home is a logistic nightmare. I found... for me it is to join ($100) AA Cargo and become a certified known shipper. Again, this is a big part of their business, shipping perishable and oversize items. Its more expensive and comes out roughly to $2 a pound..... much cheaper than UPS or Fed Ex. It stays in a cooler when you drop it off and you pick it up at a cooler at your destination.
That's basically it from me. Other guys I've talked to do it differently... more than one way to skin a Mt Lion (sorry.... can't get Mt Lion off my mind). I've never lost one ounce of meat from field to home and have only given away some to friends in Ak and a couple pretty counter girls. Ed F