Here is the full list of all 17 RePete achievements.
Collect every gem in world 1
Beat every level in world 1
Beat every level in world 2
Beat every level in world 3
Beat every level in world 4
Collect every gem in world 2
Collect every gem in world 3
Collect every gem in world 4
Beat every level in world 1 in the time limit
Beat every level in world 2 in the time limit
Beat every level in world 3 in the time limit
Beat every level in world 4 in the time limit
Fall into the first pit in the game
Beat level 2-13 the easy way
Beat level 3-7 the easy way
Keep trying after 5 deaths in the same level
Take more than three times too long on any level