Here is the full list of all 13 Konrad's Kittens trophies - there are 9 bronze, 3 silver and 1 gold.
Clear the table in the Tea party minigame.
Go to the dark forest.
Chill out in the living room.
Visit the bathroom.
Finish the kitten tutorial.
Visit the beach with your kitty.
Cuddle the kitten in plushy mode.
Wear 4 different accessories at the same time.
Give the kitten a balloon ride.
Get a total score of 10 points.
Have 100 coins in your purse.
Get 1000 points in a mini-game.
Unlock all accessories.