I really have no idea what brought this into my head, but was there ever an origin story in Flash Gordon for Ming? Was he the hereditary ruler of Mongo? Did he stage a coup? Where did everybody’s favorite space despot come from?
I’m not sure about where within the story line Ming came from, but it’s clear to me where the idea of Ming and the planet Mongo comes from. Flash Gordaon was done, as was Brick Bradford, in imitation of Buck Rogers, who was inspired by F.P. Nowlan’s “Anthony Rogers” stories from 1928 (collected, over 40 years lateer, and published as "Armageddon 2419 A.D. ). Anthony Rogers falls asleep in the early 20th century and wakes up, Rip van Winkle-like, in the 25th, when the Orientals have taken over the Earth, Western civilization is reduced to scattered guerrilla bands, and super-science prevails. Rogers joins the Guerrillas and fights back. When they turned it into a comic strip, they made “Anthony” into the folksy “Buck” Rogers. Before him those Earthmen-involved-with-future-or-alien-enemies had been gentlemen and officers. Buck was a guy you could have a beer with. The book introduced the first use of “Zap!” as the sound of a ray gun firing.
The use of orientals as the enemy followed suggestions from writers like H.G. Wells, who noted that blacks and orientals would, in the future, not be content to be downtrodden colonists, but would become political forces themselves, mixed with “yellow peril” fears, and the fact that opponents had to come from somewhere “other”. At first Buck Rogers fought earthly orientals, as well. But as time went by, he got extraterrestrial foes, and so did his imitators. Flash Gordon succeeded even better than Buck rogers, due in no small part to the excellence of the artistry. But it’s impossible to look at “Ming” the Merciless from the planet Mongo (wherever the hell that was) and not see the traces of the fear of people who had a “Ming” dynasty and who had “Mongols” in that history as well.
Alex Raymond, as far as I know, never did an origin story for Ming. Shortly after Prince Barin is introduced, he is referred to as “the rightful ruler” and “the pretender to Ming’s throne”. In a later story, Ming is shown visiting an island cemetary for the imperial family, to bury an uncle who had tried to usurp him. It is quite a large cemetary, which would imply either a long dynastic history, or a lot of collateral branches of the family, who probably all considered themselves eligible to be emperor.
As for the other writers who have done the strip in the five decades since Raymond? I haven’t a clue. When my local paper carried the strip in the early 1970s, the stories rarely took place on Mongo.
Most of the film and television versions of the story have Barin as the rightful heir to the throne, and Ming as a usurper. Details, however, are rare.
Although in Flesh Gordon, the emperor is described as “a former botanist, who went mad after being crippled by a Penis Flytrap.” 
Cites:
Alex Raymond’s Flash Gordon, published by Checker Book Publishing Group
Volume 1: ISBN#0-9741664-3-X
Volume 2: ISBN#0-9741664-6-4
Volume 3: ISBN#1-933160-25-X
Volume 4: ISBN#1-933160-26-8
Volume 5: ISBN#1-933160-27-6