Bear
Start a race on the Cyber Field 2 track. Collect the blue diamond between the two holes near the finish line. Complete the race and save the game. Bear will be unlocked on the character selection screen.
Ebisumaru
Unlock Bear. Play the Ganbare Dochu 1 and 2 tracks with Bear and beat the record time to unlock Ebisumaru.
King
Start a race on the Sky Bridge 2 track. Collect the Blue Bell power-up, then find the jump just before the large gap. Note the blue diamond on a platform just to the right of the jump. Jump towards the platform, using the power-up for an extra boost. Collect the blue diamond, complete the race, and save the game. King will be unlocked on the character selection screen.
Vic Viper
Start a race on the Moon Road track. Collect the Blue Bell power-up, then find the first long jump on the track. Jump and use the power-up for an extra boost to land on a long platform that has the blue diamond. Collect the blue diamond, complete the race, and save the game. Vic Viper will be unlocked on the character selection screen.