Here is the full list of all 13 Licky The Lucky Lizard Lives Again trophies - there are 9 bronze, 3 silver and 1 gold.
Collect 20 health pickups
Survive 3 burns from the lava in a single game
Achieve a score of 30,000
Achieve a score of 15,000
Grapple to 60 platforms using the manual aim
Release 10 floods
Gain 15 boosts from geysers
Die 10 times
Achieve a score of 5000 in a single combo
Destroy 80 platforms or boulders
Achieve a score of 5,000
Complete the tutorial
Get x10 multiplier