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 at the board selection screen, press Up, Up, Down, Down, Left, Right, Left, Right, B, A. This will open up the Ika-Chu, Father, J-B, and J-Arm boards.

Thanks to Revolution readers Ernie Rodriguez and Christian Slydahl!