Don't worry, this isn't going to be a post about literally turning chess into a game mechanic. This is more about how you can use chess as a model for implementing politics into a campaign. It is a fairly simple and easy way to keep track of things, while having in game events matter and reverberate over the course of time.
In a political campaign, whether the players are at the center of power or on the periphery, their network of personal connections and resources matters a great deal. Informants, vital contacts, advisers and other courtiers make-up this cluster of 'human' assets that extends the party's reach. It is easy to hand-wave or just wing it, but sometimes you want to give these things more weight so the players know that their decisions matter.
A while back I began imagining my political campaigns a bit like chess. Not a perfect chess match, it may be a bit lop-sided with varying pieces in play and more than two opponents, but essentially each side would have things like a Bishop, Rook or Queen. Each NPC in their inner circle of connections brings vital skills to the table and has agency. They may have someone who is adept at acquiring information from the local population or a character who knows how to levy troops. These matter in a political campaign and players should know they matter. If such a character dies or is incapacitated, that should have an effect they can feel.
I approach this very simply. I give each NPC in the inner circle (and here inner circle just means any NPC who is an asset to the party) one or two areas where they excel. So say the party has an adviser with a large network of informants. I might decide that they will learn of plots against them (or have a percentage chance of learning) before they happen. This is something that they can appreciate because there is a huge difference between the adviser telling them an assassin is on his way versus them finding out during the assassination itself. And if that adviser meets with misfortune later on, they'll understand why the assassin got through before they learned of his plans (maybe that is even how they learn of his demise).
This door swings both ways. Factions working against the party have these kinds of resources too. Part of playing the political game is eliminating the Bishops, Rooks and Queens from your opponent's arsenal. This is why in the movie the Godfather, one of the Turk's first moves is to kill Luca Brasi (he is probably a Rook or Queen). It doesn't have to be about killing or eliminating, it can be about influencing or inconveniencing. It can also be about putting someone in the right position to become that piece (like when King Richard IV makes Prince Edmund the Archbishop of Canterbury in The Black Adder).
Usually how this looks in real terms is I have my roster of characters for each major faction in a political campaign with their key skill or skills listed next to them. Because I like the random element, I assign percentage to the skill as well. So if the party hatches a plot against the King where they try to undermine his relationship with an important foreign ally, I would roll the percentage on his chief diplomat to see whether he can block their efforts. But if they take him out of the picture first or keep him busy with something else, the plan will have a much greater chance of success. Also, some plots might be so good they have a 100% chance or working, or they may require roleplaying to resolve if any of the party is actually present for the plot's execution.
Of course as with any model or method, you don't want it to consume the game or get in the way. There is always the temptation to fixate on the procedure and lose sight of what is actually happening in play. As long as you are mindful of that it shouldn't be an issue.
I have to say I quite like this approach. This is the sort of thing that works well for me, and I think it will work well for many other GMs too.