It depends a lot on how the king is chosen in this kingdom. I'll Assume quasi-historical eldest child/son model.
I would guess that if the desired king has no connections to the kingdom, you are going to have to lie, fake documents and all sorts of similar things.
If the current king has no legitimate heirs then faking that the desired king is an illegitimate child of the ruler (or rulers parent) could work.
If the king has heirs, then they are going to have to be killed or discredited first. Typically somewhere in history there is a questionable succession, evidence that someone else had a better claim (and that that claim now rests with the pretender).
Typically though these type of claims aren't resolved in any sort of court, but instead on the battlefield. If the kingdom has something like a house of lords it might be able to be resolved there though. If the lords have enough power that they can choose whoever they want (without worrying about bloodlines etc.), then just pressuring them directly could be enough, of course in that situation the king isn't likely to have too much power anyway.
See the War of the Roses for how this sort of thing works out.
Hint, it never ends without violence (unless one side is unable to field an army). In which case their is only violence against the claimant who can't support an army.
Kingdom? Arrange a marriage with a princess then kill the king + any heirs preceding the new prince(prefferably in a terrible accident). As a bonus the succubus now has perfect blackmail material on the new king ;P
How to Force a Marriage
1. Money. As seen in Aladin people with enough money can claim royalty without opposition and therefore marry into actual royalty
2. Succubus marries the king. Whispers of advice
3. Killing off the male heirs in a male dominant society
4. Threat to the kingdom. A "wiseman" approaches the king with a prophesy
Killing People
1. Bandit attack
2. Jousting/Sparring/Archery/Other sport malfunction
3. Falling
4. Drowning
5. Ill trained pet
6. Disease
7. Fatal Quest
8. Scorned Lover Murder Suicide
How about the Mind Swap, Major spell?
Swap the minds of the old ruler and the new ruler. The populace will not know that the swap occurred and now the new guy is in charge, in the old guy's body.
Major mind swap can only be cast by an 18th level psychic, such a high level caster is incredibly rare.
They probably wouldn't be involved in such a small place if its only two towns and a fort.
Scroll of Mindswap Major then. Also Ninjaed
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1) Succubus gets close with current king, close enough to regularly use Suggestion. Pillowtalk is good for this, people tend to be more willing to just auto-fail their will save :P
2) Succubus gets current king to appoint new guy as an advisor.
3) New guy has some big public successes. Becomes an important advisor.
4) Succubus gets current king to appoint the new guy to be the second in command.
5) Succubus engineers a crisis of some kind. In which the king dies. New guy takes over "to maintain stability".
Done.