Looking for Dorans in County Down

I am slightly confused. If your ancestor’s surname was Doran, why was his father’s King? Is that a mistake?

Doran is a pretty common surname in Co Down. In the 1901 census there were 573 in the county, most of whom were RC.

The RosDavies site has quite a few Doran’s on it, all over the county:

https://freepages.rootsweb.com/~rosdavies/genealogy/SURNAMES/D/Doran.htm

I’ll assume Thomas was RC. Not all Irish RC parishes have records for the 1830s and so it may be that there is no record of his baptism nor of his parents marriage. Without some idea of where in the county Thomas came from you are facing a needle in a haystack. What was his father's occupation (according to the Australian marriage certificate)? That might help.

Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. If you have already tested your DNA with Ancestry, My Heritage or 23&Me you can upload your DNA results to Family Tree DNA for free and then join the North of Ireland Family History Society DNA Project. Simple instructions on how to do this can be found here:

https://www.nifhs.org/dna/uploading-your-dna/

I too, like Elwyn I am a bit confused about the names. 
trying to help, the birth was in 1839 so the names will be in the Tithe records.......here goes..........

Thomas King ....... In the Tithe records...........
The Tithe Applotment Books, 1823-37 (nationalarchives.ie)

The townland is Derryneill
Derryneill Townland, Co. Down (townlands.ie)

If Catholic then look in here
Catholic Parish Registers at the NLI
There are two Parishes for this area Drumgooland....Upper and Lower. I have looked and for August 1839 I didn't see Thomas Dorans. 
You may have to look at Parishes on the joining Parishes.

The name Doran in County Down is this list
The Tithe Applotment Books, 1823-37 (nationalarchives.ie)

Have a look through these links and I will watch your post, ok
Margot