The First Rugby Club
The generally accepted date of formation for various clubs is faithfully reproduced below, but they have not all been verified. Many people have asked "What was the first Rugby club ?". Before answering such a question there are others that need to be considered (Some possible answers have been provided below, but no doubt other historians might have different views):
1) What is a Club? A group of persons organized for a social, literary, athletic, political, or other purpose.
2) What is a Rugby Club? An athletic club organized to play football according to Rugby rules/laws. Schools do not qualify as clubs, e.g. Marlborough school (1850) where Rugby rules were adopted but without hacking, however, former pupil clubs would qualify.
3) What Rules/Laws should apply? Some type of general adoption of Rugby Rules/Laws is required, (as was generally the case when an Old Rugbeian introduced the rules to a school or club). There was, for example, a football club founded in Edinburgh in 1824 by John Hope which lasted 10 years but this is too early to be associated with either Rugby or Association football since at that time there were no written Rugby or Association rules. Note: later other clubs published rules e.g. Blackheath 1862 
4) What if the club had some local variations? Rugby rules were documented at Rugby School Aug 28th, 1845 but these were far from exhaustive and many clubs instituted local variations see laws. Often captains would meet prior to games to thrash out which rules would apply, and then also arbitrate during the game itself . So this particular issue is open to a wide range of interpretations.
5) From what point is the Rugby Club officially founded/formed? When it holds its first general meeting to formalize the constitution of the club.
6) Doesn't the first documented game count as when the club was founded? Whether against itself (which was often the case) or against an opponent, no, since many clubs played a number of informal games prior to the club becoming official.
7) Does it still need to be in existence? No, however, if the question was phrased "Which of today's rugby clubs is the oldest?" then obviously yes. The next problem would be "what if it had a break in service for some period e.g. during the world wars?"
Considering the above and the scarce information available on which to make a judgment, identifying the earliest Rugby club is extremely difficult. The candidates commonly brought forward for the title of the earliest club include :
Barnes Club (Barnes RFC) formed 1839 but not well documented, 1858, 1862 (first recorded result).
Guy's hospital (1843), 1866 (The football annual 1873) 
Cambridge University 1846  (stemmed from an earlier football club formed in 1839 by Old Rugbeians)
Marlborough school - Rugby rules but without hacking 1850, 1852 
Durham Grammer School circa 1850
Neuenheim College in Heidelberg 1850
Trinity College (Dublin University Football club) - Dublin 1854
Liverpool [first game 1857]
Academical Football Club (Edinburgh) 1857  1858 -based upon first AGM 
Blackheath 1858 (closed club formed, Old Blackheathen Football club), 1862
University of St Andrews RFC 1858
Merchiston school 1860 
Manchester RFC 1860
Richmond 1862 (founded in 1861 but switched to Rugby rules a year later)
Club History Links
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1. Marshall, R., Football - The Rugby Union Game. 2nd ed. 1893: Cassell & Co Ltd. 564.
2. Richards, H., A game for hooligans : the history of rugby union. 2007, Edinburgh: Mainstream. 317 p.  p. of plates.
3. Thorburn, S., The history of Scottish rugby. 1980, Lond.,: Cassell Ltd. 346.
4. Scottish Rugby Union. and A.M.C. Thorburn, The Scottish Rugby Union : official history. 1985, Edinburgh: The Union in association with Collins. xviii, 210 p,  p of plates.
5. Owen, O.L., The history of the Rugby Football Union. 1955, London: Published for the Rugby Football Union by Playfair Books. 368 p.
6. Edinburgh Academical Football Club., The Edinburgh Academical Football Club centenary history : a history of the club and of football at the Edinburgh Academy. 1958, Edinburgh: The Club. 165 p.,  leaves of plates.
7. Van Esbeck, E., One hundred years of Irish rugby : the official history of the Irish Rugby Football Union. 1974, Dublin: Gill and Macmillan. 246 p.,  leaves of plates.
8. Birley, D., Sport and the making of Britain. International studies in the history of sport. 1993, Manchester: Manchester University Press. 358 p.
9. Tyson, D., London's Oldest Rugby Clubs. 2008, LLG Publishing. 168 p.