Federal law very generally allows an adult non-felon citizen to transport a firearm across state lines under 18 U.S. Code § 926A, provided he or she can legally possess it at both origin and destination:
Notwithstanding any other provision of any law or any rule or regulation of a State or any political subdivision thereof, any person who is not otherwise prohibited by this chapter from transporting, shipping, or receiving a firearm shall be entitled to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry such firearm to any other place where he may lawfully possess and carry such firearm if, during such transportation the firearm is unloaded, and neither the firearm nor any ammunition being transported is readily accessible or is directly accessible from the passenger compartment of such transporting vehicle: Provided, That in the case of a vehicle without a compartment separate from the driver’s compartment the firearm or ammunition shall be contained in a locked container other than the glove compartment or console.
This law has been on the books since 1986. There may be additional federal, state, or local restrictions on what you can transport and how, however.
Your situation here is relatively simple. Alaska is among the most gun-friendly of states, with no permits for open or concealed carry of handguns, and no license or registration required, for most people over the age of 21. If you legally possess a firearm in Washington state, you are most likely able to take it with you in Alaska. Of course, you are still subject to the applicable state, federal, and local restrictions that do exist; for example, carrying firearms in national parks was legalized not long ago, but you cannot bring them to a courthouse, school, domestic violence or sexual assault shelter, child care facility, or bar.
Obviously, TSA rules prohibit carrying a firearm, firearm parts, or ammunition inside the passenger cabin of a commercial airplane. An overview of requirements is provided on the Transporting Firearms and Ammunition page, as well as a blog post entitled Traveling With Firearms and Ammunition on Commercial Aircraft. Some rules include
- Declare each firearm each time you present it for transport as checked baggage. Ask your airline about limitations or fees that may apply.
- Firearms must be unloaded and locked in a hard-sided container and transported as checked baggage only. As defined by 49 CFR 1540.5 a loaded firearm has a live round of ammunition, or any component thereof, in the chamber or cylinder or in a magazine inserted in the firearm. Only the passenger should retain the key or combination to the lock unless TSA personnel request the key to open the firearm container to ensure compliance with TSA regulations. You may use any brand or type of lock to secure your firearm case, including TSA-recognized locks.
- Firearm parts, including magazines, clips, bolts and firing pins, are prohibited in carry-on baggage, but may be transported in checked baggage.
The major U.S. airlines will accept firearms in checked luggage on most domestic flights. subject to their own rules. You do need to declare that your bag contains firearms, firearms parts, or ammunition when you check it with the airline. When you reclaim the bag at your destination, you may be required to receive it from an agent after showing identification.
Each airline may stipulate its own requirements; see, e.g. Alaska Airlines' Transporting Firearms page. A few other airline policies are linked from Flying by Commercial Airline Scheduled Flight at GunLawsByState.com (GLBS).