It wouldn't be much different from the way corporations are run now
@AlexP is right, you normally can't just buy the office of king because there is no "super-king" to sell it to you (and if there is, that would probably be feudalism, not plutocracy). You need to look at how societies form in general, specifically thinking what would cause a group of human beings to form a plutocracy. Likely they would be forming a plutocracy from either an oligarchic/monarchic society or a republic/democratic one, as a plutocracy the way you describe would probably need a functional system of hard currency, which wouldn't be present in a group of hunter-gatherers just becoming sedentary for the first time.
We already have examples of how a plutocratic society is run in the present day: corporations. Corporations start from pretty much nothing and form when individuals decide to pool their resources together to accomplish a broader goal, not that different from how societies and their governments form. In corporations things are kind of a weird mix between democracy and monarchy, with the person who can own or control the majority of shares dictating what happens to the company. A functional plutocracy wouldn't be that much different from a cyberpunk corpocracy, though it would probably be more open to participation as works like Shadowrun and Cyberpunk 2077 treat the corporations as shadowy pseudo-governmental overlords instead of examining how corporations actually run (i.e., regular people are never shown buying corporate stock, there is never any shake-ups in leadership due to changes in who owns the majority share, and the corporations are never beholden to their shareholders, which is kind of the point of a corporation).
In such a plutocracy, citizenship would be based by the number of shares in the country you own, as in owning >1 share makes you a citizen. Owning many shares would give your vote more power, but it wouldn't make you more or less of a citizen. From a political science point of view, this would mean that everyone that "bought into" the country therefore has an interest in seeing the country succeed. This is a kind of Heinlein-esque view of society, namely that only those willing to invest in society should be allowed to influence its policy (In practice this doesn't work, as can be seen in the early 20th century when the automotive barons bought up the railroads and deliberately ran them into the ground to make their cost-inefficient cars look better. Because they were so wealthy they could eat the loss).
Public positions would probably be determined by appointment by whoever is in power at the moment. "Buying" public positions is probably not a long-term feasible strategy, for the simple reason that it will never result in the most competent people getting the post (this is arguably true for politicians today, but you'll notice that public service jobs like police officers and EMTs require skill checks, and even public sector jobs require interviews and job skills). It really seems like it's only the "elected" jobs where any old idiot can get into them. As a result, most jobs would probably be determined by appointment, similar to the modern U.S. spoils system or how a lot of corporations work. In practice a lot of people would get jobs based on political connections and kickbacks creating a "good old boys" system, kind of how politics works today, keeping the plutocracy angle that you want.