Instructions
Play through and win at least third place in the Shell and Banana Cups. This unlocks the Leaf Cup.
Go to the Cup selection screen. Choose "Leaf Cup" and "150cc."
Come in first in the Leaf Cup's tracks: DS Desert Hills, N64 DK's Jungle Parkway, GBA Bowser Castle 3 and GCN Mario Circuit. Upon completing the 150cc Leaf Cup, the Royal Racer kart is unlocked.