The Duchess of Gloucester is the widow of Thomas of Woodstock, the Duke of Gloucester. The first and only time we see her on stage, the Duchess tries to convince her brother-in-law, John of Gaunt, to avenge her husband's death (2.1). But Gaunt refuses, because King Richard is responsible, and Gaunt thinks it's more important for him to be loyal to the king than his own flesh and blood. Shakespeare uses the Duchess' request as a way to show the difference between the way the men and women of the play view family matters. For the women, family ties seem to always come first. But for many of the men, politics takes precedence over everything else.
The Duchess' grief over her husband's death, along with her insistence that Gaunt find some way to get justice, really captures the mood of the kingdom. If the king (the person you go to when you want justice done) has become unjust, what the heck are you supposed to do? Gaunt tells the Duchess that she can't do anything except wait for God to punish Richard (1.2.2). Feeling hopeless and grief-stricken, the Duchess says, "Farewell, Old Gaunt. Thy sometimes [former] brother's wife/ With her companion grief must end her life" (1.3.3). Later, we learn from a Servingman that the Duchess has died (2.2.2).