How does the ranking system work?
There isn't a traditional ranking system. Your main 'leveling up' indicators are how many weeks you've survived (shown next to your name in the lobby) and how many members you have in your Clan.
A quote taken from EvangM:
"There is no such thing as "levels" in TLOU MP. One of the things we wanted to move away from was the traditional progression iconography and descriptions that people use to make snap judgements on whether or not a teammate/opponent is 'good' or 'bad'. Even though it works well for a lot of games, like Uncharted 3 MP, we felt it was too static for a game like this that can vary so wildly from one match to the next."