If you want the full details, I would recommend just reading the LN. This is during volume 6, and if you're unable to buy the official translation, there is a fan translation that you could read right now.
But basically, Clayman's going to act all smug, declaring that Rimuru killed one of his subordinates and is proclaiming to be a demon lord. Someone (I think it was Rimuru, but I can't exactly remember) suggests that they fight each other right there. Guy agrees, creating a dome-like barrier for Clayman and Rimuru to fight inside as to not disrupt the others. Fighting commences, and Rimuru ends up having to fight Milim. After fighting for a little bit, he confirms that he can't win and has Veldora fight her instead via the Storm Dragon Summon skill. Rimuru then focuses on Clayman, they duke it out, until Clayman can't go on anymore. Lion man is revealed to be Karion, Milim was never being controlled to begin with (she had to deliberately lower her defenses to make the device even have a slight effect on her) and Frey was never his ally in the first place. The three laugh at Clayman's ignorance, then Rimuru kills him. Karion and Frey step down from being Demon Lords and instead choose to serve under Milim. None of the other Demon Lords have any problems with Rimuru becoming a Demon Lord, making 8 of them in total. The Demon Lords agonize over what the new name from them will be since it took them a very long last time and there are no longer "Ten Pillars". Veldora mentions that Rimuru is good at coming up with names, and Rimuru (after being "persuaded") comes up with "The Octagram".
On the Clayman Army vs Tempest/Yuurazania side, Clayman's army loses. Milim's followers "fight" against Tempest/Yuurazania (i.e. sparring matches) until Clayman remotely forces Yamza to become a vessel for Charybdis via a fragment of Charybdis Yamza had on him. Milim's bunch quickly heal everyone in anticipation of fighting Charybdis, but Benimaru quickly kills it (and Yamza in the process).