Race him in Candy`s Requests, level 3, mission 24.
Deluxe Barrel DK
Complete all of Candy's Challenges.
Diamond Cup
Complete Sapphire Cup on Rookie.
Expert difficulty
Complete Platinum Cup on Pro.
Funky Kong
Complete Diamond on Rookie.
Kalypso
Complete Sapphire Cup on Rookie.
King K. Rool
Race him in Candy`s Requests, level 4, mission 31.
Klump
To unlock Klump, complete the second level of the sixteenth mission.
Lanky Kong
Complete Topaz on Rookie.
Platinum Cup
Complete Diamond Cup on Rookie.
Sapphire Cup
Complete Topaz Cup on Rookie.
Selection Cup
Complete Diamond Cup on Rookie.
Tiny Kong
Complete Saphire on Rookie.
Unlock Pro difficulty
Complete Sapphire Cup on Rookie.
Unlocking characters
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.
Wrinkly Kong
Finish the mission in level 4 challenges where you must help Wrinkly Kong win the race.
