Who wants to live forever? Obviously, not Dr. Hayflick, and his research is the cornerstone of most longevity research. And if you don't know what the Hayflick Limit is, you'll definitely want to learn that for this story.
Here's an interview with him (grokked from Jay Lake's link salad). There's lots of good SF material in that short interview. Plenty of plot lines and lead points for further research.
If you read the comments to the article you'll discover our mass obsession with our own immortality. There's plenty of "teh stupid" (which has been on the increase lately) in those comments. And plenty of hackneyed SF tropes being tossed about like they were reality (or just around the corner, which they are not).
Mix in to this the issues Jay Lake has been facing with his own possible recurrence of cancer and my own issues with cancer. Also the pathology of that disease.
And then it comes, to live forever there's a simple formula. We must become cancer. We would need to harness some of the mutative properties of cancer, remove the harmful side-effects, and channel the growth within norms (or we'd become lumpy masses of flesh like the "horrible abomination that was the Dixie Chicks" from Bender's Game). Would you accept eternal life if it came at that price (you become the disease)? How would you solve the standard problems that would come with such extended life spans (just look at what's happened since the 19th Century and our limited life expansion)? What would happen to religion as we become near immortal? All the societal changes that would need to happen to handle a population that never dies (except for accident or major disease), hell just what would it do to marriage and other social contracts (vampire stories have dealt with this slightly, but only from the "limited population lives forever" aspect).
For me, I would write this story with the premise of a serial murderer running rampant. It might be this person has created and is spreading a disease, or they could be doing it the old fashioned way. Would the people treat them as a hero (like the old time gangsters)? Would they hunt the murder down like a rabid dog (because they are adjusting the status quo)? Would the benefit of serial murder to society be greater than the impact of the deaths? What if the murder is being committed by curing the "cancerness" of the people?
You're welcome to use this angle as well.