Statutory death registration didn’t start in Ireland till 1864 so you won’t find a death certificate for Charles. I assume the Durnin family were RC. Few RC parishes kept burial records, so there’s probably no record of his burial either, unless he happens to have a gravestone. (Somehow that doesn’t seem likely if he was killed at a seaport on the day of departure. Possibly he isn’t even buried in Fermanagh).
The family presumably married in the 1820s or 1830s. There’s about 18 RC parishes in Fermanagh but only 1 has records for that era (Enniskillen). The others didn’t start keeping marriage or baptism records till the 1840s or later. So there may be no church record of the family in Ireland.
Looking at the 1901 census of Fermanagh there were 55 people named Durnin (and variant spellings) in the county. All were RC. There were 659 Elliotts, of which only 65 were RC. Most were Church of Ireland. Could this have been a mixed marriage? In which case the marriage might possibly have been in the Church of Ireland. That would involve a search of all the Church of Ireland records for the county. There’s 22 C of I parishes, though again not all have records for the period you are looking at:
http://www.ballynagarrick.net/ulsterancestors/Parishes%20of%20County%20…
There are copies of all the surviving Church of Ireland records for Fermanagh in PRONI (the public record office) in Belfast but a personal visit is required to view them. They are not on-line there. Rootsireland (a subscription site) may have some of the records. It might be worth searching it’s records, if you take out a short subscription.
Though most Irish censuses pre 1901 have been destroyed, a few fragments survive, including a small part of the 1821 Fermanagh census. There was a Charles Durnin (father also Charles) in it, but he was born c 1815 and seems to young to be your family:
http://www.census.nationalarchives.ie/pages/1821/Fermanagh/Derryvullan/Lisnarrick_Village/3/
The surname is reasonably common in Fermanagh, being spelled Durnian, Durnion, Durning and Durnan as well as Durnin.
DNA testing may be the only way of matching up with family in Ireland today or with others who know more about the family’s exact origins.
