Unlike Cards, the King cannot be directly upgraded to increase its Level.
Instead, the King gains experience points (XP) each time a Card is upgraded. The number of points gained is shown by pressing the ‘Upgrade’ button when a Card is ready to be upgraded. Even Cards not currently in your Deck or not often used will gain XP for your King when upgraded.
The amount of XP required for the King to reach the next level is indicated by the progress bar toward the upper-left of the ‘Your setup’ section.