We have no indication of the manner of Heli's death.
As to Heli being the father of Joseph, this is probably not the case. Matthew gives Joseph's genealogy and states that the father of Joseph is Jacob not Heli. Most scholars concur that Joseph in Luke's genealogy is in fact the "father-in-law" of Heli. That would make this Mary's genealogy and not Joseph's. This makes sense in the record: Matthew gives the "official" genealogy and traces it back to David while Luke gives Mary's genealogy - the real one - which also traces back to David. None of the Lord's worst enemies ever denied his descent from David. This descent allowed him to claim, as Gabriel put it, "The throne of his father David."
I hope you have found this helpful.