Nan's portrait features in the engraving The Cockpit, which gives a strong clue as to her unsavoury occupation. The cockpit was a venue for cock-fighting, and Nan was a cock-breeder and well-known figure on the fighting circuit. As her nicknames suggest, she was based in Deptford.
There doesn't seem to be much more information available about Nan Rawlings. It's perhaps not surprising: although (as Hogarth shows) people of all classes attended cock fights, those who made their livings from the activity were not likely to feature in many histories. In fact, she may have been forgotten fairly soon after her death: by 1803, the 'Duchess of Deptford' was a title accorded to a lavishly-dressed figure in a print satirising the nouveau riche.
In 1835, cock-fighting was banned by the Cruelty to Animals Act. One suspects that Hogarth would have approved: The Cockpit is a depiction of the vice and degradation of its gambling audience, while his series Four Stages of Cruelty begins with its central character delighting in such animal suffering and ends with his executed body being dissected at Surgeons' Hall.
The Museum of London website has the image and a description.
Hogarth's London continues at the Cartoon Museum, Little Russell Street, until 18 January 2015 and is well worth a visit. I attended with London Historians.