Knack 2 has 3 Coliseum Attack Challenges. Getting 5 stars in all coliseum trials unlocks the "King of the Coliseum" trophy.

Coliseum Attack is unlocked after beating the story for the first time. It can then be found in the main menu under "Extra Content". The more enemies you kill without being hit, the more points you earn. 5 stars is the best score.
Make sure you kill all enemies very quickly without being hit by them! I recommend using the jump kick (X+O) and Somersault Kick (O+Triangle). Just spam this combo over and over again. It's super quick and kills most enemies instantly. The Somersault Kick needs to be unlocked in the power-up menu first (Way of Strength skill tree).

Press: X+O , O+Triangle, X+O.

Against enemies with electricity shields you should use boomerangs (hold Triangle). It disables their shield. You can also use the boomerang to stun other enemies.

If you have trouble, unlock the crystal knacks first. I highly recommend Saber Knack (Topaz Yellow).
Saber Knack deals more damage, has better defense and recovers quicker. Compared to normal knack he can kill enemies in one less hit (can one-hit many enemies). You unlock the crystal knacks by opening treasure chests.

TIMELINE:

Coliseum Attack #1 - 0:05
Coliseum Attack #2 - 1:18
Coliseum Attack #3 - 3:03