"Proper" means "correct". "The proper chain of command" means "the official rules for who to ask for help".

Use the word "proper" to contrast with doing things in a messy, unprofessional, or untraditional way. For example, if you just eat a piece of pastry for breakfast, someone might tell you:

You need to eat a proper breakfast.

A "proper breakfast" might be something like eggs, toast, juice, and bacon in the U.S.