He lived and died as a surgeon and apothecary in Lichfield; a Scottish university conferred on him, it is said, the degree of M.D., but though highly gratified he never assumed the title of doctor.
He deposited these curiosities in the ancient registry office of the bishops of that see, which stood nearly opposite the south door of the cathedral, and has long since been pulled down.
His second wife was Theodosia Webb of Croxall in Derbyshire, who died at Lichfield on 1 August 1793; she had one son, Thomas, a lieutenant and surgeon in the Stafford militia.
Greene's portrait, with the motto, described by Boswell "truly characteristical of his disposition, Nemo sibi vivat", was engraved in his lifetime, and is inserted in Shaw's 'Staffordshire,' i.
It also contained numerous curiosities from the South Sea Islands, which had been given by David Samwell, surgeon of the Discovery, to Miss Seward, who transferred them to Greene, and thus enabled him to obtain a medal struck off by the Royal Society in honour of Captain Cook.
Next year Bullock bought for a hundred and fifty guineas the arms and armour which were first exhibited at his museum in the Egyptian Hall, and were afterwards added to the collections of Sir Samuel Meyrick and in the Tower of London.
Nearly the whole of the remaining curiosities were sold for £600 to Walter Honeywood Yates of Bromsberrow Place, near Gloucester, who made many additions, and in 1801 printed a catalogue of the whole.
A woodcut from his sketch of a tombstone found in 1746 among the ruins of the friary at Lichfield appeared in its number for September 1746, p. 465; and so late in his life as 1790 he communicated to it a notice of a manual of devotion, written on vellum, and formerly belonging to Catherine Parr, the last wife of Henry VIII.
Stebbing Shaw was favoured by Greene's son with the loan of some valuable manuscripts and plates from the museum for use in his History of Staffordshire, and he embodied in his account of Lichfield a description of the collection.
When Johnson was desirous of placing an epitaph for his father, mother, and brother on the spot in the middle aisle in St. Michael's Church at Lichfield, where their bones rested, he sent the lines to Greene.