How we cite our quotes: (Act.Scene.Line)
Quote #1
KING HENRY
What title hast thou, traitor, to the crown?
Thy father was, as thou art, Duke of York;
Thy grandfather, Roger Mortimer, Earl of March.
I am the son of Henry the Fifth,
Who made the Dauphin and the French to stoop
And seized upon their towns and provinces. (1.1.107-112)
Henry believes he deserves the crown because his grandpa and dad were kings. It goes further than that, though: he also feels he owes it to these guys to hold on to that crown and keep ruling, even if he doesn't really want to. He feels he owes it to his family to keep that crown.
Quote #2
RICHARD
An oath is of no moment, being not took
Before a true and lawful magistrate
That hath authority over him that swears.
Henry had none, but did usurp the place.
Then, seeing 'twas he that made you to depose,
Your oath, my lord, is vain and frivolous. (1.2.22-27)
Richard is always one to champion loose ethics. Sure, you made an oath to Henry, he tells his dad, but oaths can be broken. Richard's speech shows us that he's willing to compromise his duties at any point if it will benefit him to do so.
Quote #3
RICHARD
Shall we go throw away our coats of steel
And wrap our bodies in black mourning gowns,
Numbering our Ave Marys with our beads?
Or shall we on the helmets of our foes
Tell our devotion with revengeful arms? (2.1.162-166)
Here, Richard talks to Warwick and Edward about what they should do with Henry. He believes that since Henry's men killed his dad (York), the brothers should fight Henry with everything they have. Even though he has a loose morality, he seems to care about his dad's legacy. He wants to make his dad proud and continue to fight, instead of just giving in. Or at least that's what he wants his brothers to think; it could be that Richard is already planning to get the crown himself and is just using his dad as way to motivate his brothers to keep the crown long enough for Richard to claim it.