Let's say you cut a guy up and spread out the pieces of his body. What would happen to him? Or is it not possible to do so in the first place (i.e. he heals instantaneously)?
I agree with @punkarsenic about someone like a cancer patient whose situation is made worse by this virus being a good idea for a main protagonist, but a really compelling protagonist should be crafted from his setting. What kind of setting and/or time period does this take place? What things do these people value the most in this society? Immortality would certainly make them more reckless sure, and they would probably try harder to pursue the things that they value. Money? Status? Or maybe even something more spiritual (like "obtaining enlightenment", in whichever way they interpret it)?
How does this virus change the the society's cultural and legal traditions? If the society used capital punishment, then that wouldn't work all that well anymore. I'd imagine prisons and the such would have a much harder time of keeping their prisoners under check.
Let's say a person assaults another. The victim likely will not have any physical proof the assault took place due to the speed at which their bodies heal (unless they happen to record the event in some way). Can the perpetrator then just easily walk free? And this situation would make torture easy-peasy for the perpetrator, wouldn't it?
Does the virus you're thinking of prevent biological growth? Can people still reproduce? Just having cells "unable to die" might create too many edge cases. Either address those situations or add further limitations to the type of immortality these people experience.
Ultimately, I think one of the big issues that the reader would expect resolved in some way would be an idea of why/how this outbreak occurred. Human experimentation--and to what purpose? Some divine purpose? What purpose can any life has when it continues eternally? A lot of human instinct evolves from and revolves around the idea of dying. What happens when you take away the core of something?
Idk if that helps, but I hope it would be somewhat useful? Best of luck!