How and when did the King instruct Santiago to use the stones?

The king you refer to in the story is Melchizedek, who also calls himself the King of Salem. Melchizedek is the one who encourages Santiago to pursue his destiny and to find his treasure.
He gives Santiago a white stone and a black stone named Urim and Thummim. Melchizedek tells Santiago that the stones will help him read the omens on his journey. Santiago must only use the stones when he cannot read the omens himself. When he is stumped, all Santiago has to do is to ask an objective question, and the stones will tell him the answer. The black stone signifies "yes," and the white stone signifies "no."
In the story, we learn that Urim and Thummim have biblical origins. In ancient Israel, the priests wore golden breastplates that contained the stones. They used Urim and Thummim to divine God's will for the future. We also learn from Santiago that he rarely uses the stones; his life and journey has always provided him with enough omens that are clear in meaning.