We definitely don't get enough Gollum action in this movie. We just see him as a shadow in Moria and a head floating in the water.
But that doesn't mean his role in The Fellowship of the Ring is unimportant. We're given a very brief overview of his past, which consists of him finding the Ring 1,500 years after it was lost by Isildur. He takes it with him to Moria where it keeps him alive, but corrupts him beyond repair:
GALADRIEl: The Ring came to the creature Gollum who took it deep within the Misty Mountains. And there it consumed him. The Ring brought to Gollum unnaturally long life. For five hundred years it poisoned his mind.
Unfortunately (or maybe fortunately) for Gollum, the Ring betrays him and falls into the hands of Bilbo. It's not clear what Gollum does over the next few decades, but we know at some point he's picked up by the enemy and tortured for information. He cracks under the pressure, screaming "Shire," and "Baggins," and sending the Nazgûl on course to Bag End.
But Gollum is set free and somehow finds his Ring again, almost as if it draws him in from across Middle-earth. Gollum in the film is reduced simply to a reminder of the destructive power of the ring, much in the way that his original name (Sméagol) is reduced just to "Gollum," the sound he makes when he coughs.
But seeing as he hasn't given up on regaining the ring yet (and also seeing as Gandalf quite plainly tells us that Gollum still has a role to play in the Ring's fate) we should probably keep an eye out for him in the sequels.