Joyce,
The events are long before statutory registration in Ireland and so you would need to rely on church records to trace Henry Finch’s family. Luckily Finch is not a very common name in Co. Down. In Griffiths Valuation in 1863 there were only 4 households altogether, and they were in the parishes of Seapatrick & Garvaghy. You haven’t said what denomination the Finch family was but if you know you might want to search the church records in those two parishes. Probably will require a trip to PRONI in Belfast.
Tradition was to marry in the bride’s church, so you might want to do the same for Clendinning. Most were in Shankill with a couple in Ballymore & Drumcree. Again probably involves a trip to PRONI. Many of the church records are not on-line so a personal visit is required to view them.
A couple of trees on Ancestry name Henry’s parents as George & Eliza.