Act 1 scene 4
Aumerle who is a cousin of both Richard and Bolingbroke as the son of the fourth son of Edward III, the Duke of York, here is showing his allegiance to Richard though he professed his friendship for Bolingbroke. In this he is like his father who does not really want to take a stand with or against either of the cousins. York sides with the one who seems strongest at the time. Aumerle seems to remain loyal to Richard throughout most of the play.
Richard is concerned that Bolingbroke may be a more popular leader that he is. Bolingbroke the Machiavellian would be aware that he would need the support of the people in any bid for the throne.
Richard's plan to tax rich men to fund his war in Ireland gets a boost when he learns that Gaunt is sickly. This line shows how little Richard really cares that his uncle is dying.