Here is the full list of all 41 KLAUS trophies - there are 22 bronze, 15 silver, 3 gold and 1 platinum.
Beat Mecha Klaus
Get the first Ending
Unlock the Arcade Mode
Relive the 6th Memory
Collect all the memories
Hack 100 CPUs
Beat The Glitch Floor
Make 10000 Jumps
Relive the 5th Memory
is that K1?
Beat The Electric Floor
Finish the K1 flight of K1 on Chapter 6 of the Electric Floor
Relive the 4th Memory
Destroy 50 Objects With Uppercuts
Beat The Office Floor
Relive the 3rd Memory
Finish the clone section of the office floor without dying
Get the hidden portal of the Chapter 5 in the World 3
Smash 20 clones with K1
Make 5000 Jumps
Beat The Ducts Floor
Relive the 2nd Memory
Finish chapter 5 of the Ducts Floor without dying
Finish the Chapter 4 of the Ducts Floor
Finish the K1 flight of K1 on Chapter 2 of the Ducts Floor
Throw Klaus with K1 for the first time!
Destroy 4 objects in a row without touching the floor
K1 joins to the journey
Beat The Basement Floor
Make 1000 Jumps
Relive the 1st Memory
Finish Chapter 4 of The Basement Floor
Solve the first puzzle
Finish a puzzle without destroying the Key
Get a memory door piece
Earn all the trophies
Finish the chapter 5 of The Electric Floor without dying
Destroy 25 Mug workers
Beat any world without dying
Beat all the records of all the Chapters
Do everything