Where in the play is Henry V at conflict with himself?

Henry V is a man who, upon ascending to the throne, has vowed to change from the feckless young man we saw in Henry IV, Parts 1 & 2. He is intent upon being "no tyrant, but a Christian king," with the result that we see him engaged in conflict with himself at various points in the play. In the opening scene, for example, he questions whether it is actually appropriate for him to claim France—"May I with right and conscience make this claim?" The Archbishop of Canterbury assures Henry that he has an excellent claim based on his lineage, but Henry remains unsure. He points out that he "fear[s] the main intendment of the Scot," given that the Scots frequently attacked the "unfurnish'd kingdom" of his great-grandfather when he dedicated his forces to France. As he struggles to be a good king, Henry must weigh numerous options and select the best, which is often not obvious to him.
It sometimes seems that Henry is not sure how to balance being a Christian with being a king; the French king notes at the end of the play, at the marriage of Henry and Kate, that "Christian-like accord" between the two countries should necessarily mean an end to war, the implication being that war is un-Christian. It is also notable that the Dauphin, in sending Henry a "treasure" of tennis balls, succeeds in stirring Henry because he thinks he is behaving like the boy we saw in the previous plays—"You savour too much of your youth." Henry's ire is aroused largely because of a desire to prove that he is no longer that boy.

And we understand him well,How he comes o'er us with our wilder days,Not measuring what use we made of them.

Henry is saying that he has grown beyond what the Dauphin thinks of him and has learned to be good: he says he will "put forth / my rightful hand in a well-hallow'd cause" and "be like a king and show my sail of greatness." The irony is, of course, that if Henry had really grown beyond his foolish youth, he would not have been so provoked by the jest of a gift. This is a tension in Henry which motivates him throughout the play: he needs to prove himself as a king, and to prove wrong others' opinions of him, but he is not always sure exactly how to do so.