Sadly the exact procedure differs by airline and airport.

By law you need a hard sided container with locks that only you can open(meaning you keep the key in your possession/don't had out the combo to anyone).

You can put a hard sided locking container with the gun in it inside a soft sided case, but that isn't very secure. Zippers can be opened with a pen, even when locked

It would be a good idea to print out your airline's firearm policy, many agents aren't familiar with them.

Also, be familiar with and print out CFR Title 49 → Subtitle B → Chapter XII → Subchapter C → Part 1540.111

Be prepared to put up a fight with the TSA. At most airports after you declare you have a firearm you will fill out a red firearm card and relock your luggage and then be escorted over to the "advanced screening area" where they will scan your bag and either approve it right then or ask you to open the bag. If they ask to have your keys to open the bag, I wouldn't put up a fight as long as I can view the entire inspection process. However, at some airports(Like MCI in Kansas City) the advanced screening area is not accessible to non-TSA or airline desk managers. This is where you should dig in your heels. I've had TSA screening managers throw fits that I didn't get my print out of 1540.111 from the TSA website because it doesn't look like their webpage(Duh, it is a legal document on another government page. However a link to it is found here.) Get their names and cards. As soon as you can contact the TSA complaint line(they are actually pretty nice, they don't like the reputation they have and seem to want to fix it). TSA employees are not allowed to refuse to get their manager if you ask, you can keep asking until you are at the highest ranking employee on site. Don't expect any of them to know the law. Throw this in their face in front of other passengers. I received applause from two guys when I told the screening manager at MCI that it is insane that I am not allowed to go to the "Secured advanced screening area" when the entire point of the TSA is to vet people into secure areas. Record it if you can.

I'm not saying to go in and be a jerk, be polite and courteous. No reason to start off all "AM I BEING DETAINED?!". But have your line in the sand and don't cross it. Use the fact that the TSA has a horrible reputation against them. Having print outs of the law is great for this, it shows how incompetent they are. Sadly their motivation for not looking stupid is your best tool for getting through if they are being stubborn.