To get your answer to the first part of this question, I would suggest starting on p. 17 of The Samurai. In case you have different page numbers, this is where we first see Velasco, though he is only called “the missionary” at first. Velasco is, at this point, in prison, where we are allowed to know his thoughts.
One reason he wants to be bishop so badly is because he wants to bring Christianity to Japan. He believes that (p.18)
the failure of the missionary effort in Japan was a result of the blunders made by the Society of Jesus, which had continually opposed his order in all things.
He feels that he could correct the mistakes that the Jesuits had made if only he were bishop. In that case, he would be able to advance the cause of Christianity in Japan. However, he also has a much less altruistic reason for wanting to become bishop. He realizes that part of why he wants to be bishop is because he wants power and he wants to be recognized as an important person. Endo says that
He realized that a measure of worldly ambition and vanity lingered in a distorted form within him.
This ambition and vanity, combined with his altruistic desire to Christianize Japan, causes Velasco to want to be the bishop.
When we first meet Velasco, it seems clear what this desire does for him. First, it has caused him to be imprisoned. It is illegal to proselytize in Japan and yet his desire to be bishop makes it so that he cannot stop doing so. This lands him in prison. Second, this desire mortifies him to some degree. He realizes that he is vain and ambitious and he does not like that aspect of himself. After Velasco is released from prison, it is harder to determine what his desire to become bishop does for him. One thing we might say is that it allows him to be manipulated by the Japanese. They know that he wants so badly to Christianize Japan that they can influence him to a significant degree (as we see Lord Shiraishi doing at the end of the chapter) by holding out the possibility that the Japanese authorities will be more lenient towards Christianity.