This can be a tricky one, but don't worry - it just needs a bit of thinking ahead.
If you are careful then you should have no problems at all with 2 people playing on the same device with different King or Facebook accounts.
What's important is that you keep these steps in mind when switching between players.
Make sure that the last person has signed out of their King or Facebook account in the game.
Force close the game.
Wait a few seconds and then reopen the game.
Finally, sign into the other King or Facebook account and off you go!
This isn't complicated but just needs the player to keep it in mind before jumping in. If you don't, you could jump into your game at a higher level, link it to the wrong King or Facebook profile and over-write the progress held there. So take a moment before you play to think about who's logged in.
If you have the Facebook app installed on the device, be sure to use the 'Login with Phone number or Email Address' option when signing into the game to avoid the game defaulting to the account linked to the Facebook app.