Double British citizen

I will be going to the UK in two weeks and I need to understand how it all works. So, I am an American citizen and a British citizen. When I leave the United States with which passport do I have to use to register and leave the United States? Then, when I arrive in the UK, how do I use my British passport? When my visit to England is over, on which passport do I have to check in and leave with to return to the United States? and finally on my arrival at LAX, how exactly do I use my US passport if I entered with my British passport at Heathrow?