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There is an excellent website for County Clare here: http://www.clarelibrary.ie/eolas/coclare/genealogy/genealog.htm
Check for any family documents such as bibles, letters, etc. Or family stories. This can reveal more information.
Do the Burkes also come from Co. Clare? Be sure to check for variant spellings of the names. Sometimes the same name can be found spelled five different ways!
You can check for information about the frequency of the name in the mid-19th century and any other variant spellings of the name here: http://www.irishtimes.com/ancestor/surname/ You can also carry out a search of the two surnames to see which part of the country they can be found together.
Civil registration records are available from the General Register Office (GRO). These start from 1864 however. You can access the website here: http://www.groireland.ie/research.htm
There are also records available for the 1901 census here: http://www.census.nationalarchives.ie/search/
There are Roman Catholic church records available for the diocese of Quin (Killaloe). These date from 1816 for baptismal records and from 1833 for marriage records. You can find these at the National Library of Ireland, Pos. 2473, or the Church of Latter Day Saints, film 0926096. If you have any difficulty, you could try writing to the parish priest for more assistance.
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64). Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at. Tithe Applotment Books (1823-38). Microfilm copies of the books for all of Ireland are available at the National Archives of Ireland (NAI) http://www.nationalarchives.ie/genealogy1/genealogy-records/tithe-applotment-books-and-the-primary-griffith-valuation/ or the Church of Latter Day Saints (LDS).
Do you know much about their emigration? Dates, the reason why they left, etc? Generally more information was given at the port of arrival rather than the port of departure. If you knew which city they arrived at (e.g. Liverpool), this could be a good place to find more information, and perhaps even find out an exact place of origin.
Hope that helps you to get started!
Please make sure you link anyone else in your family who is interested in their Irish heritage to our site - and indeed anyone else you know of Irish heritage.
Genealogist (Ireland XO)