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.
To unlock the cups below, complete the requirement listed. You can be on Rookie Mode (or higher) to grab any of them.
To unlock Cranky Kong, race his character on the third level (24th mission).
To unlock Expert Difficulty, win a Platinum Cup on Pro Difficulty.
To unlock Funky Kong as a playable character, complete Diamond Mode on the Rookie setting.
To unlock King K. Rool, race his character on the fourth level (31st mission).
To unlock Klump, complete the second level of the sixteenth mission.
To unlock Lanky Kong as a playable character, complete Topaz Mode on the Rookie setting.
To unlock a new difficulty setting called Pro, you must acquire the Sapphire Cup during gameplay.
To unlock Tiny Kong as a playable character, complete Sapphire Mode on the Rookie setting.
To unlock Wrinkly Kong, let him finish first on the fourth set of the twenty-sixth mission.