Richard II is a play rich in character studies. The most investigated character, of course, is Richard, and the heart of the play is the study of the sort of king that Richard is. The question that swirls throughout the play is, "Does the divine right of rule give a King license to mis-govern his people?" Richard was ill-suited for the position of ruler of a country, and yet, by his birth (divine right) he was the "right" King of England.
John of Gaunt is Richard's uncle and father to Richard's challenger -- Henry Bolingbroke. He serves as the King's advisor, and though he dies at the beginning of Act II, he has an important role to play in reflecting the dilemma between acting in one's "personal" interest and doing what is best for one's country. He must banish his son (who smells of revolt and treason against the rightful King Richard) in Act I, scene iii.
This decision, to set his love for his son and his opinion that Richard is not the best man to be king aside, reflects John of Gaunt's willingness to put his country first. Something that King Richard (no matter how many times he refers to himself in the royal plural "We") cannot seem to do. John of Gaunt represents the personal sacrifice, sometimes of consideration of one's own family, that one must make in true service to one's country.
John of Gaunt dies in the opening scene of Act II, but not before he proves as good as any Greek oracle, warning Richard to abandon his destructive path or lose his country. And, as might be expected of the self-involved king that he is, Richard ignores his uncle's advice and, upon his death, simply confiscates his riches for more wasteful spending. Before the play is over, Henry Bolingbroke, presented by Shakespeare as much more suitable for the role of King, will have usurped the crown.