How do you delete your clash royale account

As per Supercell's own support page, you can't delete your account.

Players can't delete their game accounts. The game servers have no access to any personal or private information about players, and no sensitive information is ever stored by Supercell.

You need to erase you phone completely because it keeps the account information on your phone unless you delete all data on your phone. Or, You could take a separate device(one that has not created a clash royale account before) and create a Clash Royale account and then transfer that to your phone through Game Center and that would do the trick.