See 1 Henry VI and 2 Henry IV for extensive background about the struggle for the throne played out here. In short, York believes that he is the rightful heir to the throne because Richard II was forcibly removed from the throne by Henry IV, so all rulers thereafter have had an illegal claim to the throne. York's lineage stems from the eldest surviving brother of Richard II, who should have been the heir when Richard II died childless. Henry, on the other hand, is both the grandchild of the usurper and a relative of a younger brother of Richard II, making his claim to the throne entirely spurious, in York's eyes.
Henry makes his claim to the throne by evoking the name of his father Henry V, who was a very popular leader and who conquered much of France, to the appreciation of the English lords. While Henry's own ineffective ruling has led to the loss of those same territories, Henry nevertheless points to Henry V as proof of the validity of his reign. Henry IV may have been a usurper, but he became king, as his heirs did in turn, and that alone should be proof of his right to power.
Yet Henry is not cut out to be a powerful ruler, and he is not enthusiastic to defend his reign. Striking a deal with York to keep the throne during his lifetime but pass it to York thereafter, Henry effectively disowns his son by eliminating Prince Edward's right to become the next ruler. As Henry gets weaker, his French-born wife Margaret gets stronger and determines to set things right through the power of the armies, now at her command.
Margaret points out the essential problem with Henry's arrangement with York: by disowning his son, Henry has become an unnatural father and disrupted the normal pattern of familial relations and monarchal succession. His weakness has made the relationships of the royal family into a relative monstrosity, where a king gives up the crown but a queen leads the armies to recover it. Margaret is repeatedly accused of being unnatural, as a woman who takes up the sword and commands armies. That surely originates from Henry's denial of his own role as protector of his son, a task that Margaret must take up.
The severing of bonds between father and son recurs in this play in several spots. In one case, Henry watches two consecutive soldiers stripping the spoils of war from the bodies of men they have killed, discovering they have killed their own father and son, respectively. At York's own death, the camaraderie of his sons is destroyed and replaced with fraternal rivalry. When kinship ties are broken, all that remains is the will of the individual, which is most strikingly demonstrated in the awesome willpower of Richard, York's son, who eventually destroys everyone else in the way along his path to the throne.