Gollum was an old, lonely creature that lived on an island in the middle of a lake deep inside the Misty Mountains. Small and thin, he was still very strong and dangerous. He ate the fish from the lake, and an unwary Goblin or two that came too close.

Where is Gollum's Cave?

The cave was located in the depths of the Misty Mountains near Goblin-town. It was close to the Great Goblin's cavern by following an estimated thirty miles of long, dark, and narrow tunnel.

Did Gollum live in Moria?

That was probably in the autumn of the year; after which all trace of him was lost. (Gollum survives, dangerously, in Moria for a while, exploring, stealing food from Orcs and evading capture by them.)