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Race him in Candy`s Requests, level 3, mission 24.

Complete all of Candy's Challenges.

To unlock Klump, complete the second level of the sixteenth mission.

Generally, when you're unlocking characters (by playing a "cup" challenge), you can predict the type of character you will unlock. If you play as a "Kong" character (like Donkey Kong or Diddy Kong), you will unlock a like type character, while if you play as a Kremling (like Kritter or Kip), you will unlock a character similar to them.

Finish the mission in level 4 challenges where you must help Wrinkly Kong win the race.