This can be attained after completing the 7th level: "Attack on Cloud Cuckoo Land".
Turn around and head over the Rainbow Bridge into the Colosseum. Walk over to one of the yellow circle pads and press (B).
Select the first (or any) character and press (A) on the '? option' to randomise the custom character and unlock the achievement / trophy.
*Note: This was done in Story Mode but it can also be done in Free Play mode.