Get over 40,000 points in "Street Work" mode.

Complete one of the following task to unlock a bonus character:

Earn an "A" rank on each level and track in "Street Work" mode.
Earn an "S" rank on each level and track in "Street Work" mode.
Complete each level and track in "Time Attack" mode within the time limit.
Earn a "Perfect" rank on each level and track in "Coin" mode.

Unlock all four bonus characters. Then at the board selection screen, press Up(2), Down(2), Left, Right, Left, Right, B, A. Now the following four bonus boards will become available:

Ika-Chu (Double jumps)
Father (Longer air time)
J-B (Press A + B for turbo jump)
J-Arm: (Extra turbos)