Statutory records of births started in Ireland in 1864. Protestant and non conformist marriages 1845, RC marriages 1864. So the events you are interested in all pre-date that. You'll be heavily reliant on church records, if they still exist, to trace this family. Really you need to know which religion they were and which parish or townland they came from to trace them.
The names are both very common. There are 368 Black entries and 83 Linton entries in Co Londonderry alone, in the 1901 census, so without some way of narrowing the field, it's a needle in a haystack.
Elwyn