Exton only appears in a couple of scenes at the end of the play, so it can be hard to remember who this guy is and why he matters (especially when you're expected to remember all the other players in Richard II). Here's a tip: Exton is the guy who 1) executes Richard II and then 2) gets exiled for his actions. That's easy enough to remember, right?
We know why Exton murders Richard at Pomfret Castle (King Henry hinted to him that he should do it), so let's talk about why it's so unfair (and ironic) that Henry banishes the guy from England. First of all, Exton was just following the king's orders. We're not justifying the murder, but we do think it's pretty lame that Henry never actually admits that he ordered the hit. Instead he says the following:
[...] though I did wish him [Richard] dead,
I hate the murderer, love him murdered.
The guilt of conscience take thou for thy labour,
But neither my good word nor princely favour:
With Cain go wander through shades of night,
And never show thy head by day nor light. (5.6.6)
What's so weird about this is how Henry refuses to acknowledge that he is the one who caused his cousin Richard's death. When Henry has the nerve to compare Exton to Cain (the guy who kills his brother Abel in the Book of Genesis), we're sort of speechless. If anything, it's Henry who's like Cain, so it seems unjust that Exton should be banished.
This scene also reminds us of how and why Richard banished Henry and Mowbray back in Act 1, Scene 3. Remember, when Henry publicly accused Mowbray of murdering Gloucester, he was really accusing Richard without coming out and saying so. Richard responded by throwing Henry and Mowbray out of the country. Is Henry's banishment of Exton really any different than Richard's banishment of Henry and Mowbray? Maybe Henry has more in common with Richard than he thinks.