You have to do this in Original Mode, NOT Arcade Mode.
The licenses do NOT stack, so you have to get these on separate occasions. Just do a 5 minute round for C, B and A. And a 10 minute round for S. Be careful when going for the S license. If you earn over 10,000 you will get a higher rank than S. In that case, you will NOT get the S license achievement.
Class C License - $2000.00 to $2999.99
Class B License - $3000.00 to $3999.99
Class A License - $4000.00 to $4999.99
Class S License - $5000.00 to $9,999.99