Here is the full list of all 16 King Kaiju achievements.
Catch hold of the helicopter holding the score display
Enjoy the beautiful day by eating some ice cream
Beat Level 1
Squish a person by throwing a car at them
Take down a news copter
Beat Level 2
Beat Level 3
Squeeze a large building chunk with your grip
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
Throw a car high up into the air to make your own fireworks