Said I'd write this up for the folks on discord. So here goes.
The basic idea here is to give a brief summary of the Demon Kings, their personalities, how that reflects in their actions and how this might play out in their overall arc.
Lucifer
Starting from the top, Lucifer. Lucifer is heavily inspired by the Lucifer of pop culture. Particular the likes of Milton's Paradise Lost. He's presented as this beautiful, tragic figure with a lofty ideal of Human/Demon coexistence, but in truth, this is a facade.
Lucifer claims to want this peaceful coexistence without suffering, but all evidence points to the truth, he wants an end to his suffering, and everybody else are merely tools to achieve that end. He uses the threat of force to compel Mephisto and the Order into helping him, even though this causes countless humans to suffer because of Section 13. He hands his siblings over to Satan without a thought and when those siblings die, he shows no remorse. He talks a good game, but everybody he speaks well of is a matter of utility. Homare is there to manage his pain, Shima is an idiot, but a useful idiot, Todo is one of his chosen for his experiments and so on, all of Lucifer's relationships are a matter of convenience and what the other party can provide him, proof and point, Michael Gedoin. Now, anybody who's seen the Illuminati arc can tell Gedoin was not a good man in the slightest, but as we saw, the instant he fulfilled his usefulness, he was eliminated. The Illuminati oath is one of self sacrifice to the cause, the chosen ones were sacrificed to the Elixir experiment and Homare was left behind. The only person who doesn't give up anything is Lucifer. Even giving up his body for the experiments, its all in the pursuit of his gain.
Shie said it best, Lucifer uses pretty words to cover up a fundamental contradiction, in his world of equality, he's the one with the power to get what he wants and I think that will lead us to seeing him being the final villain of the series despite Satan being more powerful.
Mephisto
Mephisto is absolutely fascinating, cause I still don't know what his game is. He's slimy, conniving and duplicitous, he knows more than he lets on, and leaves the heavy lifting to others when he can avoid it. Despite being the second strongest amongst the Demon Kings, his style is definitely that of a manipulator, and even then, he's a shockingly up front one. Everybody and their mother knows he's sketchy and he basically does nothing to hide it. Taking what he says at face value, he is on the side of humanity, but only because humanity provides him entertainment and distraction. If something came along that better served his hedonistic tendencies, he'd probably side against humanity just as easily as he sided with them. Even his acts of kindness seem to merely be calculated moves in a game only he can see.
Whilst he obviously takes heavy influence from Goethe's Faust, he is much more than that, seemingly being the one character that might actually be aware he's in a manga. While he might appear lazy and unhelpful on the surface, it's clear that he's the one holding this circus together and that every inch clawed back in the fight against armageddon, is at least in part due to his machinations.
As for where this might lead him, I don't think Mephisto is likely to selflessly sacrifice himself for humanity, but I think there's good odds his body might meet its end in the upcoming arc as a calculated cost of victory.
Azazel
Azazel at the minute is dubious, we know virtually nothing about them. They sided with humanity and became a founder of the Order before returning to nature, but it's unclear if that might be something they regret. Lewin questioned if Azazel was truly on humanity's side, and while, over time we can probably say yes, what, if anything currently motivates them, is a mystery.
I suspect we might find out more about Azazel and their goals since current arcs are focusing on Lewin, but I've been wrong before.
Iblis
Iblis is the first of the Demon Kings I would say is entirely based around denial. Iblis is vain and self-centred, violent and petulant to a fault, but at its core, it's her denial of reality that motivates it all. In her early existence, she was powerful and beautiful, she had a child whom she was presumably proud of, but had to watch grow old and rather than watch them die, she killed him herself. This insistence of her own power and beauty is simply a way to run from what she fears, the march of time and admitting that she fears losing those she loves and herself to the inescapable passage of time. This is highlighted in her conflict with Lucy Yang, a human who lived incredibly long, but did not fear the end like Iblis did, creating contrast between the two. She also holds the dubious distinction of being the first Demon King to die outright.
Egyn
Egyn is the other Demon King of denial. Unlike Iblis, who runs from her own mortality, Egyn knows what's going on, but is lying to herself about what it means. The scientific brains behind the Illuminati, she's tried hard to drink the kool-aid and believe that what Lucifer was doing would be a general benefit. Egyn has basically made it her mission to not piss off her siblings by being overly agreeable and doing what they tell her, in particular Lucifer and Iblis, but only when confronted by Beelzebub about how doing so ends up hurting others, does she start to express herself.
Unfortunately, I think this new found confidence might result in Egyn getting light nuked by Lucifer, as we know his tendency to discard those of no use to him.
Astaroth
Astaroth is absolutely fascinating to me, because he's not particularly complex like the others, he is in fact, hilariously straight forward, but it makes him weirdly endearing in a way. Astaroth is effectively, a living parent complex. He's so desperate for the approval of his father figure (Either his own or his host's) that he's willing to do some seriously heinous stuff in pursuit of that end. He does eventually come to realise that he's seeking the approval of people that are fundamentally terrible, but its only when he possesses Rin that he starts to realise what he's done. He finds that, despite everything that's happened, Rin remains optimistic and compassionate and he envies that in a way. In an attempt to reconcile, he plans to kill himself, but Rin stops him, officially turning Astaroth to their side, which is absolutely mental for the first villain of the entire series.
Whether or not he'll survive to truly turn over a new leaf, I've no idea, but I guess we'll see.
Amaimon
Amaimon was the first Demon King introduced to us by name and title (Astaroth had some dubious canonicity for a while, don't worry about it) As such, he's truly demonic in nature. He' destructive, lacking in empathy and generally moves to the beat of his own drum. The only reason he is seemingly 'Neutral' amongst the Demon Kings is because he's forced to abide his covenant with Shemihaza when called. Unlike the others, I would say that Amaimon's problem is that he is lacking in humanity. While all the others, despite their demonic nature, still suffer fundamentally human flaws, Amaimon is detached, having few things that bring his life any joy or meaning and as such pursues some greater sensation, though with decidedly different interests from his elder brother Mephisto.
Having already bailed from the plot, I don't see why Amaimon would return for the big fight, but I can't imagine we've seen the last of him either.
Beelzebub
This lil goober. The last Demon King introduced, Beel is cute and all, but also pretty selfish. It's pretty clear their motivations are all about their own interests, but unlike the others, their interests are thankfully fairly benign. They do seem to lack a bit of empathy, showing no concern when Bael is destroyed by Shima, though perhaps this is because Bael already had another body waiting, even then, I can't imagine it was fun for Bael. Very child-like in disposition, Beel seems to not be the brightest, but also far from the most belligerent of their siblings, so at the very least they're not likely to turn against humanity, but also might crash the BBQ market on principle.
I don't know if Beel will survive, but I think their odds are better than Egyn and Astaroth.
All in all, I hope this was entertaining for somebody.