The Duke of York is Henry's cousin. The thing to know about him is that he dies during the Battle of Agincourt. In the play, Shakespeare holds up York's death as an example of chivalry and honor. In fact, the news of his death moves King Henry and Exeter to tears.
Here's how: In Act 4, Scene 6, we learn that York was mortally wounded on the battlefield when he spotted his cousin's (the Duke of Suffolk's) bloodied body nearby. Exeter explains how York lay down beside him and cried out, "Tarry, cousin Suffolk. / My soul shall thine keep company to heaven [...] We kept together in our chivalry" (4.6.15-16; 19). Pretty dramatic, don't you think? It's also very moving because it demonstrates just how loyal York is to his fellow soldier and cousin. Earlier in the play, we heard King Henry refer to his fellow soldiers as his "band of brothers" (4.3.62). York's death is horrific and tragic, but it also exemplifies the play's notion that warfare forges a fraternal bond between soldiers who fight together.