how the elite four are chosen

Whenever an existing Elite Four member dies or retires, they usually have someone (or a few different people) in mind to take their place, and they submit the idea to the rest of the Elite Four and the Champion for approval. If it's agreed upon that person is hand-picked to be the next Elite Four member.

If the person dies/retires without picking a successor, the rest of the Elite Four and the Champion hold a meeting to decide on someone.

Usually it's highly skilled trainers--former champions, particularly powerful challengers in the past who didn't quite make it to champion for some reason or another, gym leaders, Frontier Brains, etc. I think the only hard and fast requirement would be that you need eight badges, or if you don't have someone who DOES have eight badges vouch for you.

Also when you first become an Elite Four member (and you're not Unovan) you start at the bottom, as the first member, and whoever was the first member gets promoted to second.