According to some sources, Salvemini earned a doctorate in jurisprudence in Pisa in 1729 - according to other sources, he studied mathematics there, which also seems more compatible with his later career, although he may have actually done both.1
In 1751 he became a lecturer in Mathematics and Astronomy at Utrecht (1751-). Doctorate from Utrecht (1754), Professor at Utrecht (1755-1764), Rector of the University of Utrecht (1758-1764). Royal Astronomer at the Berlin Observatory (1765-?). Fellow of the Royal Society of London (1753).
Boswell supped with Castillon on October 1, 1763. He mentions him a few times in his letters and themes throughout his Holland stay but doesn't mention any other meetings with him (which doesn't suggest anything at all, as the journal of his stay in Holland sadly went missing already in Boswell's own time).
On July 9, 1764, Boswell visited the Professor in Berlin, and they appear to have met and talked occasionally in the following months.
On September 26, 1764, Boswell, then "a happy fellow" visiting the Court of Dessau, wrote in his journal:
"Where are all my gloomy speculations at Utrecht, when I imagined that I knew all the circumstances that could arrive in human life, and that the result was only insipidity? Castillon gave me no bad answer to this. "You know," said he, "all the circumstances of human life, as you know the ingredients of which a dish may be made; but in neither of the cases can you know what will be the effect of a selection and mixture, till you try." (Source: Boswell on the Grand Tour , vol 1, p. 106)
Salvemini's Discours sur l'origine de l'Inégalité parmi les Hommes can sometimes be found via the [Abebooks] used books search engine. Search for Salvemini da Castiglione, or the actual title.