Online access to LDS is available through www.familysearch.com
. It's a great site, full of good info, but sometimes you have to take it with a big grain of salt.
I went thru my notes. I have a John Bourke listed as having arrived in Boston c 1715-16 as per Ancestry databases. I don't know if this ties into the family or not.
LDS shows John Bourke b c1600 Ireland m. Catherine Fay Mountcashel b 1603.
From there it skips to Richard Bourke b 1664 Kill, County Kildare, Ireland, d 1727. (They have Richard listed as the s/o John Catherine but she would have been 61 yrs old! I think there's a generation missing here.)
Richard Bourke was the father of John Bourke b c1690 Co. Kildare, Ireland, d 1762 VA.
John was the father of James Burke b c1710 Ulster N. Ireland, d 1783 Surry Co. NC. James married Mary Jane Bane in 1730. (This is my line, through his daughter Mary.)
There are many flaws in this lineup, esp. dates. But it's a good starting point.
After looking at LDS, try Ancestry's family trees; they are free and you can download them. James Burke is listed there as well. In fact he's all over the place! Lots of descendants. Hope this is of some help...