How do I find Biographical information on the 19th century English
painter Williams Gibbons?

lerak,
No entity matching your description found.

A famous English woodcarver, Grinling Gibbons (1648-1721) is a
possible match, however I feel it more likely you are referring to
William Gibbons Preston, an American architect also known for his
paintings.

The possibility exists as the Internet generally refers to Mr Preston
using first, middle & last name. In examples though, the emphasis has
been placed on last (then first) name leading to possible confusion in
name (i.e. Preston, William Gibbons (1)).

If this is the party you are interested in only a brief
bibliographical note has been found, with many links to his
architectural work. For further information you may consider reposting
question after considering comments made and reflecting on pricing of
question.

Reference:
http://whitemountainart.com/Biographies/bio_wgp.htm
Links:
http://www.boston.com/news/local/articles/2004/12/19/our_french_connections_are_legion?pg=2
http://attractionguide.com/savannah/
http://www.bostonhistory.org/m_downtn.php

(1) Use browser find button, with word ?William? etc 
http://www.nationalregisterofhistoricplaces.com/RI/Washington/state.html
http://www.nationalregisterofhistoricplaces.com/MA/Plymouth/state2.html

Kind regards,
gfmaster

Dear lerak

William Gibbons has been described as a "Plymouth artist", and he
painted in Devon and Cornwall.

The 1881 census has information which is presumably about him:

Birth Year      1841 
Birthplace      Exeter, Devon, England 
Age             40 
Occupation      Artist (Painter) 
Marital Status  Married   	  	
Dwelling        35 Tavistock Place
Census Place    Plymouth, Charles The Martyr, Devon, England 
http://www.familysearch.org/Eng/Search/frameset_search.asp

You could try asking the Plymouth Art Gallery:
http://www.plymouthmuseum.gov.uk/main/information/index.cfm?fsa=dspEnquiries

In 2003 they held an exhibition which included Gibbons' "Smeaton's Tower":

"Maritime Paintings (throughout 2003) is a collection of artworks
depicting life on the ocean wave including works by William Gibbons .
. ."
http://www.bbc.co.uk/devon/culture/2003/museums_galleries.shtml

These auctioneers have handled some of his work:
http://www.plymouthauctions.co.uk/about.htm
http://www.paintingsandclocks.com/paintings/watercolourpaintings.htm

I hope you can get more help by contacting local experts.

Best wishes - Leli

More on William Gibbons (1841-1886) as referred to by leli-ga:
http://www.antiquesandfineart.com/dealers/item.cfm?id=546&pid=6129&pg=10

His household in 1881:
http://www.familysearch.org/eng/Search/census/household_record.asp?HOUSEHOLD_CODE=1881BR_2223665