Moore is a very common surname in Ireland, and so very hard to research if you don’t know exactly where the family originated. The 1901 Irish census is the oldest complete census in Ireland. There were 684 Moores in Co. Tyrone, 34 of whom were named Robert and 49 William. The names would have been even more common in the mid 1800s as the population was considerably greater then. (It was 8 million in 1841 and it’s only 6 million today).
You really need to try and find Robert’s mother’s name (from a marriage or death certificate perhaps) to be sure of tracing the right family. It would also help if you can say what denomination William was. Birth registration didn’t start in Ireland till 1864 and 1845 for marriages. Robert’s baptism and his parent’s marriage are well before those dates and so you need to rely on church records to trace the family. To do that we need the denomination and an approximate area to search.
There’s 42 parishes in the county, and probably 250 – 300 churches. Not all the churches have records back to the mid 1800s and of those that do, many are not on-line. Only the RC records are fairly comprehensively on-line. For all other denominations it’s very patchy. A lot have been copied and are in PRONI in Belfast, but there are some small churches where the Minister still has the only copy of the records. No-one has copied them at all, and the only way of checking them is to contact him/her. Searching all the various church records for Tyrone is a mammoth task. But even then, there’s no getting away from the fact that the Church of Ireland lost a significant portion of its records in the 1922 fire, and that other denominations didn’t always keep records, or if they did, they have been lost or damaged.
The only hint I can give you comes from the middle name Burleigh. That’s not a common name in Tyrone. However baptism records for the 1830s rarely contained middle names and so you can’t rely on finding that. He was probably baptised as plain Robert Moore. Having said that, Burleigh is probably a name connected with the family. (Presbyterians sometimes used the mother’s maiden name as a middle name for one or more children). There’s just 1 Burleigh family in Tyrone in the 1901 census. They were a family with a drapers shop in Dungannon and were Methodists. However I can’t find any baptisms or deaths for that family in the Dungannon area 1864 – 1901, and wonder if they had moved there from somewhere else. The surname is much more common in the adjacent county of Fermanagh, and I wonder if they originated there (though Tyrone is given as their birthplace in the census).
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. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project and can offer FTDNA testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).