The king is a ruler.  He is a conscientious yet ambitious.  He has determined that if he knows the answers to the three questions,

"...he would never fail in anything he might undertake." (paragraph 1)

These three questions become extremely important to him. When the wise men of the kingdom all give him different answers, he thought about the answers they had given him and disagreed with them. He was intelligent and open-minded.  He continued his search, so you could say he was persistent. 

He decides to visit an old hermit in the mountains who would only interact with the common folk.  He was open-minded enough to believe that this man, who was poor and lived alone, might have the answers to his three questions.  He also was not too proud to dress as a simple man, discarding his bodyguards, to visit the old man.

When he reaches the old man, he noticed that the old man was frail and breathed heavily while he was spading his garden.  He, again, was not too proud to take the spade and start digging in the garden.  He also was kind in trying to help the man. 

He showed that he was caring when the man came out of the woods with the wound in his stomach. 

"....the King again and again removed the bandage soaked with warm blood, and washed and rebandaged the wound." (paragraph 20)

When the man asked for water, he fetched it himself. 

However, there is another side of the man that Tolstoy does not really let us know.  The man he had helped, the man with the wound in his stomach, was the king's sworn enemy because

"....you executed his brother and seized his property." (paragraph 23)

The reader does not know the reason for these actions and can only judge the King on what he is doing at the present.  Again, he shows his compassionate side and a sense of righteousness.  He says he would

".....send his own physician to attend him and promised to restore his property."

By doing the three actions that he did: helping the old man spade his garden, attending to the wound of his enemy, and making peace with his enemy, he learned the answer to his three questions.