What are the indirect and direct characterization of the two characters: Courtier and semi-barbaric king in The Lady, or the Tiger?
In "The Lady, or the Tiger?" there really are only three major characters. The princess is the primary character, of course, but she and her actions are clearly influenced by the two characters you mention in your question--the king (her father) and the courtier (her lover). Each of them is clearly characterized by both the description of the narrator (direct) as well as their actions (indirect).
The king is, indeed, semi-barbaric. We know that because the narrator describes him this way in direct characterization:
He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts.... When every member of his domestic and political systems moved smoothly in its appointed course, his nature was bland and genial; but, whenever there was a little hitch, and some of his orbs got out of their orbits, he was blander and more genial still, for nothing pleased him so much as to make the crooked straight and crush down uneven places.
Indirectly, we know he is semi-barbaric because he creates this spectacularly unjust and vicious form of "justice," because he blandly sentences the man his daughter loves to this court of justice, and because he is able to sit and watch without feeling the consequences of his actions.
The courtier is, the narrator says in direct characterization, a perfect example of a classic romantic hero; and it's not surprising the princess falls for him.
[He] was a young man of that fineness of blood and lowness of station common to the conventional heroes of romance who love royal maidens. This royal maiden was well satisfied with her lover, for he was handsome and brave to a degree unsurpassed in all this kingdom....
Indirectly, we know that he dared to love a princess and that he was not surprised to get caught and be placed in the arena for justice. The best indicator of his character (which comes through indirect characterization) is the fact that he knew the princess would find out which door held the tiger and which held the fairest maiden in the land--and was willing to allow her to decide his fate. That is a stunning display of trust; and, given the princess's semi-barbaric tendencies, he has to know it's possible this will be his last act on earth.
The reason the reader truly does not know which door he will open is due to the courtier as much as the princess. We know the war which is raging in her between despair at losing her lover and awful jealousy at the thought of losing him to another woman. What is going though the courtier's mind is hidden from us except for one clear fact--he knows she will have the secret of the doors. It's his decisive move to the door she indicates (again, indirect characterization) which leaves us in such suspense.