I'm reading Nick Bostrom's Superintelligence (which is in the Audible sale now for $5) and in chapter 6 he explores your question. His argument is that a superintelligent AI, one that has recursively optimized itself, is smart enough to devise a plan to take over the world. It is assumed that that intelligence is also socially a superintelligence. It could bribe, convince, blackmail people, organizations or countries into doing whatever it wants done to achieve its goal.
Basically, a superintelligent AI is so smart that it can develop technologies and strategies so advanced that it can work. Writing this down I realize how hand-wavy that sounds - but I do think that we can likely not imagine or understand an AI of such superintelligence, so its methodology will always be a vague guess.