He appears to have had some kind of professional falling out with Shakespeare's company around the time of the premiere of HENRY V. Interestingly, that play makes a point of killing off Falstaff, a role we believe Kemp made famous. In HAMLET, which comes out a bit later, the clown is notably dead (though there is a funny gravedigger), and there's also a long lecture to a troupe of actors about the evils of improvising onstage for a cheap laugh when a play is being presented.
He may well have gotten on people's nerves.
Having been fired, or something, from Shakespeare's company, Kemp dreamed up a publicity stunt (since imitated) as a kind of comeback: dancing his way from London to Norwich. The gimmick ate up nine days on the calendar, and gave rise to the expression "nine days' wonder."
The comeback was a failure. He died poor and obscure.