You're indicating a possessive-the implied missing word is 'home' [or house]. So to turn the sentence around it would read: Christmas at the home of the Kings. This means that you have two choices: Christmas at the Kings' or: Christmas at the Kings's. Be guided by how you would speak of the home - would you say: This is the Kings' house, or: This is the Kings's house? Either is correct.
Forgive the misuse of colons in my anwer. I didn't want to confuse matters by putting quote marks around the phrases!
Been thinking about this for most of day, on + off,....but now I've just read ael's suggestion I've spluttered my tonic water all over the keyboard!! Excellent answer :D
The noun is implied [the home OF the Kings] therefore must be observed. If you omit the apostrophe, you are saying: Christmas at the more than one King.