Here is the full list of all 11 Lizard Lady vs the Cats trophies - there are 9 bronze and 2 gold.
Clear first level without taking any damage
Clear second level without taking any damage
Clear third level without taking any damage
Clear fourth level without taking any damage
Clear entire game in one go without taking damage
die
Clear the final level without taking any damage
Score 100 in Nightmare 1
Score 100 in nightmare 2
Score 100 in nightmare 3
Score a cumulative score of 1000