What we know:
The Black Rose was an extremely prominent and powerful secret society of mages of Noxus from before Darkwill took over and established the military and High Command as the absolute rulers of Noxus.
Even in those extremely early years the Matron of the Black Rose was a woman named Leblanc (see notes during Sion relaunch event)
This Leblanc provided Darkwill with early examples of necromancy, resulting in Sion's current state of madness and undeath
Eventually, Darkwill and the High Command took over. While most of these societies were either destroyed or fled/were exiled (Grey Order?)...the Black Rose simply faded into obscurity. But in truth the group still met in secret.
Years later we know that Swain and Evaine were both members
Popular Theories:
Leblanc is not a single person, but a title and false name (Emilia Leblanc) given to the current Matron of the Rose (Evaine), with the headpiece that is passed down providing an illusion to cause the wearer to appear as the original Leblanc (see LB's judgement).
Swain and Leblanc then worked to overthrow Darkwill, placing Swain on the throne as Grand General and allowing the Black Rose back into its former position as the power behind the throne.
The current endgame is unclear. Swain's lore gave him the expressed goal of ruling Noxus, while Leblanc's was more about returning the Black Rose to power. Both of those goals are currently completed. What's next could be anything:
-Molding Noxus into what they believe to be best
-Start expanding, seeking to extend their rule over all of Valoran, destroy Demacia (no League in current Lore to oppose them)
-focus on removing any last traces of Noxian opposition (Riven, Boram Darkwills eldest son Draythe, Katarina, Grey Order?)
-turn against each other, right now Swain and Leblanc effectively share power. The military is still the official ruling body of Noxus, of which he is the supreme leader, but much of his success can be attributed to the Black Rose, of which Leblanc is the leader. Neither can rule absolutely while the other is still around
-do nothing but sit around having fancy tea parties