I'd like the same voice actor to play both the black, and white kings. 

I'd imagine the white king to sound calm, and regal, think along the lines of Asgore from Undertale. And I'd like the black king to still sound regal, but with a bit more menace to his voice. If you want undertale comparisons, think of the King from Deltarune.

Here's some ideas on what to sing:
-Music of the Night - Phantom of the Opera