King Melchizedek appears to Santiago to help him achieve his destiny. He tells Santiago that lots of people give up on their destinies because they become distracted or too cautious. The king says that he often appears, in different forms to different people, to try to help them realize their destinies, and that "most of the time people don't realize" that he has helped them. The reader might infer here that the king would like more of the people he has helped to know and appreciate that it is him who has helped them, and this might be an indication that the king's flaw is perhaps his pride.
Later in the story, after helping Santiago, King Melchizedek reflects that he will probably never see him again. The king also thinks, somewhat despondently, that Santiago is "quickly going to forget [his] name," and this makes the king feel a little sad. He wishes that Santiago would one day tell somebody else about him. He regrets that he didn't repeat his name to Santiago, so that one day, when Santiago perhaps might speak about him, he "would say that I am Melchizedek, the king of Salem." The implication here is that King Melchizedek is proud, and wants to be recognized and praised for the good deeds he has done.
King Melchizedek recognizes this flaw of pride in himself. Indeed, after indulging in the thoughts outlined above, he looks to the skies and, addressing God, says, "I know it's the vanity of vanities … But an old king sometimes has to take pride in himself." The fact that he recognizes this flaw in himself suggests that it is not such a great or considerable flaw at all. The worst kind of pride is the kind which is so strong as to be beyond self-awareness. The king's pride, however, is a small and relatively insignificant flaw, in part because he openly recognizes it as such.