As far as his on-screen history goes, not much was explored in the movies when it came to Bossk. Nothing was said about him specifically in “Empire” and it’s only through merchandise and other supplementary materials that we learned more about him, including his name. One thing that was revealed in 1982 is that the Trandoshan is (or at least was) royalty of a sort.
In “Star Wars” #67 published by Marvel Comics and written by David Michelinie, Bossk is referred to as the Monarch of the Qotile system. While the story is largely focused on Chewbacca, C-3PO, and R2-D2, this is a fascinating little nugget that helped flesh out the character a bit more. Unfortunately, these older comic books are no longer considered canon as they are part of the old expanded universe, now dubbed “Legends” tales by Lucasfilm. But it’s certainly something that could be revisited at some point.