Here is the full list of all 16 King Kaiju achievements.
Beat Level 2
Beat Level 1
Throw a car high up into the air to make your own fireworks
Squeeze a large building chunk with your grip
Beat Level 3
Enjoy the beautiful day by eating some ice cream
Take down a news copter
Squish a person by throwing a car at them
Catch hold of the helicopter holding the score display
Grab a plane in each hand.
Balance a building so that it can have nice scenic view of the city
Build up a combo of destruction totaling at least $25,000
Beat a level as Megacat
Eat 50 trees
Crush 25 dumpsters with your grip
Toss a person up in the air and catch it in your mouth