... And away he goes, Precious! Achievement
... And away he goes, Precious!10
Defeat Gollum as Gollum.
This achievement can only be unlocked after completing 'The Secret Stairs' in story mode, which will unlock Gollum as a playable character.
To get this achievement, you'll need to make both players Gollum and have one kill the other. To do so, enter any level in free play and press and hold and select the fifth circle going clockwise. From here you can select ANY character. Select Gollum, repeat this process with the second player, and kill one with the other.
If you wish to unlock this achievement in Middle Earth rather than in a story chapter in free play, you will need to wait until you've completed all of the story chapters, since free play selection of characters is not available in Middle Earth until then.