Esther,
I did a bit more research. Looking at the statutory birth records (which start in 1864) I found a Thomas John Feeny born 24.6.1867 in Moss-side to William Feeny and Martha McMaster. Perhaps that’s the same William & Martha Feeney that you are looking for? Could her surname have been McMaster rather than Smith?
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1867/03477/2277455.pdf
Also Samuel 8.1.1865 & James 18.1.1871 to the same parents. James seems to have died in 1971 aged 0.
William Feeney married Martha McMaster 19.1.1850 in Macosquin (Co. Derry). He was 21 and a servant, she 22. His father was James, a labourer. His address was Drimcroon, hers Camus. Her father was Archibald McMaster, a farmer.
There was no birth registration in Ireland prior to 1864. You need to rely on church records for baptism events before that year. To do that you need to know what church the family attended. Based on the fact that I have found that there was one other Feeney family in the area who were Presbyterian, that may be the best denomination to start with. The nearest Presbyterian church is Moss-side. It has baptism and marriage records from 1842, plus burial records for the years 1855-1878. I don’t think these records are on-line anywhere but there is a copy in PRONI (the public record office) in Belfast. If you can’t get there yourself, you may need to get a researcher to look them up for you. Researchers in the PRONI area: http://sgni.net
I had a look for deaths for William & Martha in Ireland but didn’t see any. That suggests they had left the area. So the next obvious place to look was Scotland because that’s where the vast majority of people from Antrim and nearby counties went to, to look for work.
William & Martha are in the 1881 Scottish census, at 11 Bouverie St, Port Glasgow, with children Eliza Jane 18, Samuel 16 & John 13. Possibly their departure from Ireland coincided with William’s departure for Australia. William, Eliza Jane & John were all mill workers in Port Glasgow, their son Samuel was an apprentice engineer. I don;t know what became of Thomas John but given that he would only have been 14 in 1881, I'd guess he had died.
I can see a death for Martha Feeney or McMaster in Greenock West in 1908, aged 78. GROS ref 564/1 348. That’ll tell you whether William was still alive then, and you’ll know which side of 1908 to look for his death.
You can also look for the family in the censuses 1891-1911 in Scotland. Plus you may be able to find some marriages for the 3 children that went to Scotland. Assuming this is your family, you’ll maybe have living relatives in Scotland today. (For example, I see a marriage in Port Glasgow in 1885 between Samuel Feeney and Rosann McClure. That looks promising. I can see Samuel & Rose Ann in the 1901 census for Port Glasgow with 5 children. And so on. There’s a 1904 marriage in Port Glasgow for John Feeney to Elizabeth McKean. That might be worth checking too).
The births, deaths & marriages and censuses are on the Scotlandspeople site. You need to join and buy a few credits to see the records:
https://www.scotlandspeople.gov.uk
I also found this Presbyterian Feeney family in Moss-side in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Drumtullagh/Mosside/922886/
http://www.census.nationalarchives.ie/pages/1901/Antrim/Drumtullagh/Mosside/922892/
David John Feeney married Margaret Jane Watten on 27.1.1888 in Drumtullagh Church of Ireland. David’s father was Robert Feeney (who was born in Co Derry according to the 1901 census, so not originally from the Moss-side area).
Robert Feeney married Hannah McCallum on 22.1.1855 at Roseyards Presbyterian church. Robert’s father was also named Robert. Both were labourers.
Can’t see any immediate connection with your William born 1852, other than Moss-side itself. But they may be connected. Possibly William born 1829 & Robert b c 1835 were brothers.