I concur with the other posters that you shouldn't make the whole faction die just because the King died. What if the King gamethrows or is just a bad player in general? What if they do everything right, but a TI happens to check them N1? Having the whole faction die because one player dies is unfair and too punishing. My
Chessmen faction had a system where if the King dies all members are roleblocked for the night. That's punishing enough where you want to protect the King but not so bad that you automatically lose the game because of it. I would recommend doing something similar but a little different for your faction.
I like the wincon, but it feels like the faction is built more around being an evil faction than a neutral faction. It's balanced around 1 KPN, which is too much when you also have an NK on your side. Even though it's a little strong as it is, I would suggest changing the killing mechanic because people don't like not being able to do anything. You should also reduce the killing power somehow since you do have an NK on your team. I would aim for something in the 0.3-0.5 KPN. To accomplish this, don't have the Champion kill someone each night, and instead add some sort of cooldown mechanic or other restriction.
Now here are my thoughts about specific roles:
King - What if there is no non-MR role? Overall having two players to decide the kill will cause this faction to have the same issue as Mafia, where two players are needed for the nightly kill and there is one less opportunity for a faction member to use their normal night ability. Maybe have the King attack their target, but if they're roleblocked or something then the Champion does the kill. And King should be able to choose a Champion from jail.
Queen - Maybe give this some sort of other ability? I don't have any good ideas atm but you should make Queen similar to King but with a certain strength over the King and a different weakness.
Heir - Probably doesn't need to exist if you change the promotion system, so maybe you can make it a regular support role. I would recommend giving it an active ability, especially since it's quite a bit weaker than Heiress right now. Also I would suggest removing the kill since surviving the attack is punishing enough for the attacker.
Heiress - Fine as is, but if you change the promotion this should also be a support role.
Royal Guard - This might work fine in AA, but it's going to be very weak in Ranked where there aren't many attackers. Maybe protection should also involve changing their investigative results?
Peasant - Like the previous posters said, this encourages gamethrowing since the Peasant can out the King and confirm themselves as Surv. Especially if you change the promotion system I'm not sure how this role would work. You should probably either remove it or completely change it.
Stalker - I don't like the idea of giving people access to private chats since players will just choose not to talk. Maybe let it see its target's whispers instead?
Cavalry - This faction seems to be balanced around having 1 KPN, and having other role that can kill could be swingy and unfair. Even a single kill can have a large effect on the game, so you shouldn't mix killing and non-killing roles. Instead of killing the player, I would suggest having some other effect if the target visits the Cavalry, but it would have to be quite strong. As a buff, maybe have Cavalry deliver the effect if the target visits any Medieval player.
Beheader - Generally single-use abilities aren't too popular. Instead, maybe have the Beheader select someone each night, and if they get voted up they get lynched immediately.
Actor - This is fine, but I would recommend removing the notification for players who are invited to the show. It doesn't really matter, but it would allow the Actor to hide from Lookout better and maybe allow Medievals to pretend their target wasn't home.
Preacher - What happens if you select a Vigilante? Are they roleblocked or do they attack you? Having the killers attack you seems a little unfair since if you visit a non-killing evil role you gain information but if you visit a killing evil role you die. You could make this a regular roleblocking role, but that would be similar to Heiress.
Overall the faction isn't bad. Some roles need to be rebalanced and the mechanics need to be fleshed out, but I can see the faction working overall.