Slim. There are 30 major league teams, but hundreds of coaches and former players looking to get into the coaching thing. That’s why your resume of success is paramount. Your best bet is to have success as a player, finish your career with the same team you started with, become an assistant coach as soon as you retire, become great friends with owner, GM, and current manager, get voted into the Hall of Fame, and stick around until the current manager retires or gets fired. Then you will be the natural successor. Easier said than done.
If you are determined to make it as a major league manager, you might have to work your way up the ladder by coaching in the minors, getting a job as an assistant, schmoozing anyone who will listen, and waiting for your opportunity. When you do get it, don’t waste it.