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The 1841 UK Census suggests he was born in London around 1791.
He was granted freedom by redemption in the Patternmakers Company in 1816. The application shows his trade as a jeweller.
He was assessed to Land Tax on property in Bishopsgate Street from 1818-1843; he entered his mark at Goldsmiths Hall from number 171 in 1828.
The 1820 London Poll Book listed him as a silversmith and the one for 1830 as a jeweller.
He was recorded as an unmarried silversmith aged 50 years in the 1841 UK Census records for the same address.
The 1845 Land Tax assessment book, showing him as the proprietor of a shop in Bishopsgate Street, is the last record found for him.

All sources noted so far record the address as 71, rather than 171, Bishopsgate Street.

Some further detail of Charles Gibson can be found here: Gibson's Medicinal Spoon

Recorded as Jeweller and Silversmith of 71, Bishopsgate Street Within, Cornhill.

Source: Johnstone's London Commercial Guide - Corrected to 31st August 1817 - Andrew Johnstone - 1818

DISSOLUTION OF PARTNERSHIP

Gibson, C. and Hopkins, A.B., Bishopsgate-street Within, goldsmiths

Source: European Magazine and London Review - May 1821

Trev.