Hello! First time DM with my game starting when school starts next week. 

My campaign is themed around civility as a mask for (mostly institutional) violence. The BBEG is the lich vizier(/interpreter) to the holy king of this holy city. The king is a massive leviathan beast with ornate ivory armor giving it the appearance of a human face. People treat it like a rational actor and revere it as a God. It slobbers and scratches at the walls and eats its handlers and the interpreter makes some kind of meaning out of its senseless violence. The creature-king is magically confined to its labyrinthine palace complex, and also acts as the vizier's phylactery (when it eats people). 

The main conflict is that the vizier is close to breaking the seals keeping the king trapped in this palace.

I need help figuring out how the villain can break the seals! Something villainous on its own to raise the stakes. Do they require blood sacrifice? Or maybe involving a magic substance that turns people frothing and violent. Ritual deaths of those who placed the seals in the first place? Maybe the king doesn't start in armor, but his armor is built over the course of the campaign from the horns/teeth/tusks of great beasts, which, when complete, would allow him to exit the palace. But then how do my players stop him!

Would appreciate any notes very much, thank you for reading :].

So if the king represents institutional violence, I think the mask/armor not only protecs the king, but also restrains it. The king needs the armor removed to escape the palace and go on its rampage. How does the armor come off? That's a good question, but I think ultimately it's about belief. The people the king "protects" (notably not all the people, just enough of them) need to want the armor to come off. Perhaps the mask/armor itself is sustained by belief in the way that some fiction portrays gods.

Okay, so why would the people want the king to be released? There's probably a few ways you could approach this, which is good because planning one single path through your campaign that requires the villain to succeed in certain specific ways is a trap you must avoid. One way this could happen, though, is for the lich vizier to enthrall a large segment of the population, perhaps using an artifact that your PCs could try to destroy. Or the vizier could do it the old fashioned way and simply convince people that they'll be safer and more prosperous with the king unmasked.

How can the players defeat the king once it's out of the armor? I think there's probably at least two ways, which leads to an interesting final choice. Do the PCs bind the king in a new or rejuvenated set of armor, halting the immediate problem but ultimately returning to the status quo? Or do they kill the now vulnerable king, shattering the status quo but becoming responsible for breaking the state monopoly on violence? (This is kind of a strained allegory, but don't get me started on the problems with running allegorical D&D campaigns--we'll be here all day.)

Going with the theme of civility masking greater violence, maybe the seals themselves are spread to the four corners of the kingdom to remote areas, each of them emanating a radius of some detrimental effect that the local communities suffer from. Maybe one gives off an aura of plague and that town has a high rate of illnesses, while another maybe gives off an aura of sleeplessness that gives everyone horrible nightmares, etc. The party is being contracted by the vizier to help ease the suffering of these towns and end the curses, BUT what was forgotten was that way back when the king was originally sealed, these curses were entered into willingly to fuel the spell that sealed him away. This has passed out of living memory for all but maybe a handful of old townsfolk (who try to spread the word about the necessity of the price they continue to pay, but ultimately people think it's just mystic babble) and the Vizier. The vizier presents this as a mission of relief to the players; a decent thing to do for his most long-suffering of subjects, but in reality this civil act would unleash the greater violence that lurks within. In this case, the road to hell is literally paved by the party's good intentions. 

Maybe as the players break each seal, a piece of the King's ivory armor shatters, revealing more and more of the horrid beast.

Sorry this isn't strictly related to your question but I did want to throw this in, since you're a first-time DM: running a campaign that requires the villain to nearly succeed at the end is one of the most common pitfalls of homebrew games. It's understandable; this is how many linear stories, such as movies and books, create tension. But a story like this has very high potential to go off the rails in a TRPG setting, where the players (ideally) have much more agency in the story than the reader of a book. This also has the potential to really mess up an intended allegory, because your players don't have to engage with your symbols the way you expect them to. (Example: maybe your players defeat the lich much earlier than you expect; does the king still get released, or does the point of the story accidentally become "nurturing institutional violence is fine actually as long as you kill the weird little guy who runs interference for the institution"?)

The other thing about D&D that makes these types of stories difficult is that the rules in 5e really only define two outcomes to combat: "Victory" and "Death". It's difficult to deal non-death setbacks to the party that feel fair and earned, because these setbacks typically have to be outside the players' control (e.g. you defeated the lich, but somehow the king managed to get loose anyway!) So what you end up needing to do is create a setup wherein the PCs can constantly succeed, but the villain continues to make progress regardless. This _can_ be enjoyable for players, but it's a very delicate balancing act.

Now, to be clear, I'm not saying "don't do this campaign". The campaign sounds really cool and I think you'll have a blast running it. What I'm saying is: "be very careful how you set this up." I would not clearly define specific, numeric conditions that must be met for the king to be released, eg "The five seals must be broken", because then if that doesn't happen you're dead in the water. You could maybe have the villain tell a lie like that, but you want to know it's a lie from the beginning so you can be consistent about how the seals or whatever actually function.

All this to say, it's very easy to paint yourself into a corner with an ending plan like this, and I encourage you to keep things flexible to avoid that.

each of the seals needs to be broken by bathing them in the blood of 7 good-hearted folk

This and your other comment are really fantastic notes! Thank you so much. 

The "villain must accomplish a defined numerical choke points of success/failure" was definitely throwing me off. Ideally, the campaign would focus on the players trying to prepare a counter-ritual to defeat the king or seal it away before it can get out, and I needed something for the villains to be doing in the background to raise the stakes/make them feel more active/act as a kind of doomsday clock. I was definitely overcomplicating things by getting that specific with it, and I'm really grateful for you pointing that out. 

The removal of the armor is such a smart idea. I love the imagery of getting glimpses of more and more of the monstrous form underneath the surface as violence in the world increases and it builds so much tension for when the mask comes off completely. 

This is so smart!! 

I love that if something had trapped the king inside his palace, they would have also needed to endure some kind of sacrifice/detrimental effect. One of my party members has an old god fey patron, and I love the idea of them being stuck in some kind of perpetual fruitlessness/autumn/famine for a thousand years in order to keep the king sealed away. I also love the party directly contributing to the Vizier's plans, and the armor progressively shattering. Thank you so much!!