Hi. 5th cousins is likely. I've got my DNA on 23&Me. In some ways, the DNA isn't that helpful because it just shows that someone is, in my case, 68.8% British and Irish, which I knew. (What has been interesting is the rest of my European ancestry - 4.4% French and German, and we know that in the Burke family there were several people of German ancestry).
Most of what's on the PEI site refers to William Burke as being from Northern Ireland. That narrows things down considerably. There's a electoral roll site called 192.com and it shows 200 Burkes in Northern Ireland. I suppose one step would be to write to all those Burkes and ask them if they do any genealogy research and what they've found... Did any of them have a great x X uncle named William who was in the British Militia and went to PEI?
Another thing that seems to be in the documents is that he was in the British Militia. This could be prior to joining the St. John's Volunteers/Prince Edward Island Fencibles. If that was true, then the Imperial War Museum/s (
http://www.iwm.org.uk/ ) would be a place to visit.
I would also really like to know where he was from and something about his life. That's the point of genealogy to me, the stories of the people, not just ticking off boxes in a family tree.
I've lived in the UK for 42 years and had always planned to investigate but for the past few years, work has prohibited me getting more involved in my genealogy research.
Have you seen the Muster Rolls, St. John's Volunteers 1784 on the PEI site? That shows William (and one woman and three children at that point) living at Little Rustico, Lot 34, with 100 acres. Lot 34 is less associated with our Burke family than Lot 43, but this could be a mix-up (because the Island Register doc shows him as living on Lot 43), so this may be him.
I'm sorry I can't add anything. My only "success" in the last few years has been possibly finding out who John Jamieson's wife was. It seems she may be Elizabeth Lowrie...
Can you PM me your email? I'd like to stay in touch.
Linda