Bear - To unlock Bear go to the Cyber Field 2 track. Collect the blue diamond between the two holes near the finish. When you complete the race Bear will be unlocked as a playable character.
King - To unlock King go to 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. Grab the blue diamond and complete the race. King will be unlocked as a playable character.
Vic Viper - To unlock Viv Viper go to 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. Grab it and complete the race. Vic Viper will be unlocked as a playable character.
Codes thanks to GameWinners.