Wendi,
Birth registration didn’t start in Ireland till 1864. Prior to that you need to rely on church baptism records. Not all have survived and of those that do exist, not all are on-line.
The most complete collection for Northern Ireland is in PRONI (the public record office) in Belfast. Their records are not on-line and so you need to go in person to search them (or get a researcher to do that for you). There are about 200 churches in Belfast so if you know his precise denomination that should narrow the search a little.
Looking at the 1901 census of Ireland there were 20 people named Charles Rodgers. They were a mix of Catholic, Church of Ireland, Methodist and Presbyterian. Those are perhaps the likely options for your Charles.
Can you not get his parents names from his second marriage certificate or his death certificate? Several trees on Ancestry say that Charles’ parents were Bernard 1792 - 1866 (died 15.3.1866 in PA) and Isabella 1824 – 1882 (died 5th March 1882 PA). Another tree has his father as Hugh 1825 – 1881 (died 24.7.1881 PA).
You mention the parish of Shankill as a possible place of birth. Be aware there are 2 such parishes in Northern Ireland. One in Belfast and the other near Lurgan, Co Armagh.
The best sets of passenger lists are on Ancestry and Findmypast. Have you searched them? Be aware that the biggest port of departure for North America was Liverpool. Far more Irish emigrants left from Liverpool than directly from Ireland. It acted as a clearing house for migrants from all over Europe and had daily departures whereas the numbers of sailings directly from Ireland were much smaller.
