Let's say there is a business man who owns a company (also a CEO) and moves to a wealthy municipality. He wants to be the mayor or at least the de facto undisputed ruler of the area. How would he go about this endeavor?
If he is wealthy himself, he can become a good Samaritan. Get busy using his own money to do whatever the wealthy residents would like done without having to pay for it with higher taxes. Buy or otherwise get involved with with local media so what he does gets lots of attention.
It is relatively easy for people with money to look like leaders, because indirectly, their money (in effect) hires followers by funding the infrastructure that volunteers need to operate: Office space, computers, printing and marketing costs, donuts, signs, transportation, electric bills, people to write press releases and bulletins, hiring professional artists and musicians for events, hiring security and stage builders and stage and sound technicians, hiring attorneys to secure permits and licenses and rent venues for rallies and demonstrations.
Doing something good for a community requires truckloads of money even if the focus is on volunteers. The important thing, for your protagonist, is to make sure THEY are the voice and face of all these good deeds, and nobody else. (except perhaps a spouse as a secondary).
Name recognition and positive feelings among the electorate are the two main ingredients to getting elected. To get things done the CEO needs political allies: Those can be acquired, before their own run for office, by advocating for and supporting those that will be in office when their own bid is made.
It would be important for these potential allies to have an "understanding" up front that their supporter [our CEO] will run for the top spot eventually, and is hoping for a friendly reception at that time. As a CEO, presumably they are good enough at reading people to tell if the person they are backing will reciprocate in kind, without needing to be explicitly asked. (To avoid any charges of quid pro quo.)
It might serve the CEO to support underdogs, so favors are definitely owed. Much of politics is conducted in this silent "understanding" manner so there is no possibility of a record of collusion or corruption. I do you a small favor without you asking. You then do me a small favor without me asking. I complain about X (just venting, I'm not asking for help, I don't even say I'm "hoping" for help) and you are smart enough to do something about it. And vice versa; our relationship escalates from tiny favors to large favors, whenever needed. If there is no reciprocation on the tiny favors, there is no escalation: You prove yourself to me by being smart and figuring out what I need, I prove myself to you in the same way. Even if we are secretly recorded or videotaped at any point, anywhere, even if we think we are alone, there is nothing to suggest anything criminal or corrupt at all.
That's the smart way to do it, and that is the way our CEO should operate. Money well deployed can legally buy allies and supporters, both in the civilian population and amongst other politicians.
If you are talking about a present-day, real-world, Western-style democracy, the mayor is not the "undisputed king" of the city. The mayor, city council, sheriff (if elected), and perhaps the judges acting in concert might be.
So your character has to influence all of these. That sounds as if he has to get political allies elected into various offices. Just as a regular political party does.
The character would either join an existing political party closely aligned with his past and present views (so there are not too many inconsistencies to dig up), or found a new one. The undoubted administrative skills of a CEO and plentiful private funding would come to his aid. A well-organized, well-funded campaign should get your character and his allies into the various offices.
Of course there could be a bit of a backlash against CEOs turning political leader these days. Did that happen in your setting?
Well gaining political power is about greasing the right palms to a large extent, real power might take longer. Maybe a series of hostile corporate takeovers to disrupt oil (or whatever industry dominates this municipality) and establish yourself as the hand that feeds (or starves) politicians and thereby influences their decision making.