The Duke of Austria comes to the play with a heavy backstory: he was responsible for the death of Richard the Lionheart. Even though we don't get all the details on how this happened, we get the feeling that Austria isn't a very nice guy.
Of course, when we meet him at the beginning of Act II, he is allied with King Philip of France, and King Philip is on a mission to support Arthur, Richard I's nephew. Thus, it's important for Arthur and Austria to have a public show of forgiveness between them; Austria even pledges that he will not go home until he has achieved his mission of putting Arthur on the throne of England.
Is there any chance that Austria is being sincere when he says this? Well, sincere or not, it turns out he was telling the truth: his generally belligerent personality makes it inevitable that he would end up going head-to-head with an equally belligerent personality—the Bastard, who also happens to be the son of Richard I, the man Austria killed. The Bastard also happens to be an ardent supporter of King John—one more reason for Austria to support Arthur and not John.
In the second battle between England and France in Act II, the Bastard kills Austria and walks onstage holding his head.