Gollum's backstory was already written by a great writer.
Smeagol was a hobbit who lived by a river, found the ring, was corrupted by it, and hide with it in mountainous caves. His significance to the narrative is that he was the one to find the ring after it had been laying dormant at the bottom of a river for thousands of years, presumed lost, and through him Bilbo and then Frodo acquired it. He is also significant because he represents what Bilbo or Frodo would have become if they had kept the ring for too long.
His story has been told in the films. We've seen him before he found the ring, we've seen him commit his first murder in order to secure the ring for himself, we've seen how his humanity was lost and how he was driven into the mountains, we've seen his encounter with Bilbo, we've seen him follow Frodo to Mount Doom and die in the fires. What's left of his story for us to see?