Northumberland (a.k.a. Percy) is pretty bitter under Richard's government. He's unhappy with the king's decisions and eventually becomes Henry Bolingbroke's right-hand man. In fact, he's pretty instrumental in helping Henry overthrow Richard. At one point, Richard calls Northumberland a "ladder wherewithal / the mounting Bolingbroke ascends [the] throne" (5.1.3). In other words, without Northumberland's help, Henry wouldn't be king.
In the play, Northumberland also plays the role of "bad cop," especially after Henry becomes king. (Guys like Northumberland are a dime a dozen in contemporary politics, because political leaders like to have other people do their dirty work for them.) It's really important that Henry look good while he's taking power, especially during the difficult scene where Richard hands over the crown (4.1).
Luckily for Henry, Northumberland is more than happy to do the unpleasant stuff that would make the new king look bad, like asking Richard to read over and sign documents while the former king is sitting there, crying and heartbroken. At one point, Richard tries to stall, and Northumberland says "My lord, dispatch. Read o'er these articles" (4.1.2). Translation: "Quit your boo-hooing, Richard. Hurry up and sign these papers already." Northumberland is so mean to Richard that, before he can cram the documents down Richard's throat, Henry has to tell him to ease up and leave the poor guy alone: "Urge it no more, my lord Northumberland" (4.1.14).
P.S. Northumberland plays an even bigger role in the next play, Henry IV Part 1, where he becomes disgruntled with the new leadership and leads a rebellion against King Henry. Hmm, is it just us, or are you detecting a behavioral pattern here?