Early Life
Benjamin John Johnson is currently a brick wall as I can find no sign of a baptismal record. He was noted as 71 years old when buried in 1818, making his date of birth some time between the 21
st of April 1746 and the 20
th of April 1747.
There is a number of candidate baptismal records (narrowed down to London based on where he spent his life):
- Benjamin Johnson, b. 30 Aug 1747, bapt 20 sep 1747 to James and Rachel at St Anne, Soho [on FindMyPast]
 
- John Johnson, b. 28 Feb 1746, bapt. 27 March 1747 to John and Mary In Westminster (note that this is slightly too early, unless there's an issue with Julian/Georgian dates)
 
- John Johnson, born 24 Mar 1745/6 and bapt. 16 April 1746 at St Lawrence, Jewry, to John and Mary.
 
Bill Johnson's pedigree (derived probably from handwritten pedigree by 
Walter Hugh Lowthian Johnson) notes Benjamin, born 1746, as the son of John and Mary (and the latter John is shown as the son of another John and Mary) but without any sources. This is obviously nicely consistent with either the second or third records mentioned above, but not firm enough to really clarify the relationship properly.
Contrary to this, a bit of analysis of the names of Benjamin's children (see Family Life below) might suggest that Benjamin's parents were called William Johnson and ____ Fountain.  We can certainly find a Mary Fountain who married a John Johnson, but this was in 1680 (in London) which is too old to be Benjamin's parents, but could be his grandparents.
Career
Benjamin is listed as a "Gentleman" in his probate documents, and recorded as living at 
5 Abingdon Street, Westminster (a building which no longer exists), implying he was well-to-do. There are various American Loyalist claims documents which list a Benjamin John Johnson (of Jermyn St) being an "attorney" acting for various citizens who lost assets during the American War of Independence. This would suggest Benjamin was a lawyer or barrister, which is in line with his evident wealth.  His will (see below) makes mention of a Charlotte Skinner, who can also be found in discussion of American Loyalist claims 
at this link.  There's a small chance that the Caroline Smyth also mentioned in his will is Georgiana Carolyn Pitt Pilkington, daughter of 
John Carteret Pilkington and later the partner of Joseph Brewer Palmer Smyth, an American Loyalist (see 
this entry at Wikipedia.
Family Life
Benjamin married 
Ann Paterson on the 28
th of November 1780 at 
St George Hanover Square. The wedding was witnessed by Walter Paterson (could this be the father of Ann?) and another witness whose name is unclear but looks like V Haleth.  
They had at least two children:
- William Fountain, born 1781, who married Isabella Hawthorn but had no issue
 
- Walter Rankin, born 1787, who married Louisa Stephens and had two children (Walter and Isabella)
It seems unusual that they only had two children with a six year age gap, but so far I've been unable to find records for any others.  I wonder if William, as the elder son, was named after his paternal grandfather whereas Walter was named after his maternal grandfather?  This then leaves the question of where the "Fountain" and "Rankin" middle names come from. Rankin is a Scottish name, so probably comes from Ann Paterson's side; perhaps her mother's maiden name?  Equally, "Fountain" might be Benjamin's mother's maiden name.
Death
Benjamin John Johnson died in April 1818, aged 72 and was buried at 
St Margaret's, Westminster, which would have been right round the corner from his house on 5 Abingdon Street.
Some extracts from his will:
I give and bequeath unto my dear wife 
Ann Johnson all my furniture, plate, linen, china … and other effects (not herein by me specifically bequeathed which shall be in and about my house in Abingdon Street aforesaid or such other House wherein I may happen to dwell at the time of my decease) for her own use and benefit. I give and bequeath unto my said dear wife such part of my library as she may choose to select and take to her own use; the residue of my said library I give and bequeath unto my two sons 
William Fountain Johnson and 
Walter Rankin Johnson to be equally divided between them.
I give and bequeath the Annuity of forty pounds per annum purchased by me on the life of [?] Caroline Smyth an American Loyalist payable quarterly at the Treasury and also another Annuity of thirty pounds per annum on the life of Charlotte Skinner granted as a pension to her payable every four months at the War Office.
And whereas as one of the Executors and residuary Legators of the late Davison Munton Esq. I have a claim to some Estates in the island of Jamaica I hereby give and bequeath all such interest and advantage which may hereafter arise from such claim to be equally divided between my two said sons William Fountain Johnson and Walter Rankin Johnson, their sons/heirs[?] and assigns.”
And a relevant bit from Davison Munton’s will:
“It is my particular desire that my brother Anthony Munton will deliver to Mrs Johnson of Abingdon Street the two pair of silver candlesticks[?] in my plate chest which I hereby give her as a proof of my regard for both her and Mr Johnson’s friendship and kindness.”
Davison Munton is a bit of a mystery man (though the death of his wife is mentioned in 
this document.