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

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

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

Successfully complete Street Work mode with an "S" rank in each level.

Successfully complete Street Work mode with an "A" or greater rank in each level.

A bonus character will be unlocked after completing one of the following goals:

Earn an "A" rank on each level and track in "Street Work" mode.
Earn a "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.