In order to increase your Postknight rank, you must earn tokens to prove your capabilities as a Postknight! You can earn tokens from Delivery Quests, which are available from the mailboxes located in every town:

Each Delivery Quest will have a rank displayed, which indicates the estimated difficulty for the delivery. The higher the rank, the higher the difficulty of the delivery:

Once you decide on a Delivery Quest, you can see the rewards that you can get. The higher the rank for the delivery, the better the rewards that you'll receive!

When you've earned yourself some tokens, tap on your Postknight rank button:

This menu shows your current Postknight rank, and allows you to turn in the tokens that you've earned in order to increase your Postknight rank!