Before we get started on Henry IV Part 1, let's have a little recap of what went down in the preceding play, Richard II. Remember Henry IV Part 1 is the second installment of Shakespeare's tetralogy, so it helps to know about events leading up to our play.
Super-condensed summary of events from Richard II:
In 1399, the king of England, Richard II, banishes Henry Bolingbroke (who later becomes King Henry IV) to France.
Later, when Henry's father (John of Gaunt) dies, Richard takes all of Gaunt's land. "No fair!" says Henry, who isn't about to let King Richard steal what's rightfully his.
So, Henry returns from exile with an army to reclaim his inheritance. The Percy family (Hotspur, Northumberland, Worcester) back him on this because King Richard's wrong to take land away from the nobility. Besides, Henry says all he wants is claim his rightful inheritance, which seems reasonable.
But, in 1400, Henry sees a window of opportunity and helps himself to the crown when he corners and bullies Richard after the king has returned from a trip to Ireland without protection. As the newly self-crowned king, Henry sort of orders Richard's murder after he has him locked up in prison. Henry claims he didn't and that there was a "misunderstanding" between him and the guy who did the dirty work.
In any case, King Henry feels kind of bad about having a king murdered so he vows to go on a crusade to Jerusalem so that God will forgive him for his sins.