Here is the full list of all 37 The Lord of the Rings: Gollum achievements.
You finished Chapter 10.
You saved the Cook, the Elf guards, and the Riddlemaster.
You threw 50 stones.
You killed 10 Orcs.
You finished Chapter 9.
You finished Chapter 8.
You played the entire riddle game with Mell.
You rang all the bells at the bell tower.
You finished Chapter 7.
You finished Chapter 6.
You got the Grashneg trampled, drowned, and eaten.
You finished Chapter 5.
You finished Chapter 4.
You finished Chapter 3.
You finished Chapter 2.
You fell for a distance of 40 meters and survived.
One Collectible down.
You fell into lava.
You've died for the first time in Barad-dûr.
You finished Chapter 1.
Dived a total distance of 2000 metres.
You looked at all interactibles in Gollum's cave.
…to catch a fish, so juicy-sweet.
All Gollum choices.
All Sméagol choices.
You found 20 Collectibles.
Well done. You found all those useless things.
You stole food from Mell's table.
You completed the “Halls of Grond” without taking damage.
You found all food items at the bell tower.
Turn off 10 lanterns.
You discovered all vantage points.
You completed the game without dying.
The Cook, the Elf guards, and the Riddlemaster all died.
You turned Riddlemaster's painting seven times.
You jumped from the bell towers at the waterfalls three times.
You unlocked all achievements.