Couple of things not mentioned in @inspired truth's solution:
1- It seems that the player earning the achievement must initiate the MC for this to count. To ensure this, they must hold down the B button, with the other player(s) holding down B afterwards. Once all players have the B button held down, the sequence will begin. If another player initiates the MC, it won't count towards your 10.
2- The only player who needs to have the MC is the player earning the achievement. In other words, if you're doing this locally with 2-4 controllers, you only need to earn 10x 2 Player, 3 Player and 4 Player MCs on one character - no need to unlock all MCs with all characters.