Bonus characters:
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.

Bonus boards:
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)

Hint: Getting an "S" rank:
Get over 40,000 points in "Street Work" mode.