Question

what is the way monarchs gained legitamacy for their governments

Answer 1

tradition.

Answer 2

are you asking which is the way or what is the way?

Answer 3

Well, you have it a little backwards. Usually legitimacy comes first, then the throne. That is, nobody has even a chance at ascending the throne unless he has some very strong claim to being its legitimate occupant, e.g. you are the son of the last king, God spoke to you from a burning bush with half the nation listening, et cetera.

On the rare occasion that someone crowns himself king, or is crowned king by some faction or other, without yet having legitimacy, you are usually in the middle of a civil war, invasion, et cetera. In this case, the usual way the king gains legitimacy is simply by outlasting or overpowering (and probably killing or exiling) any other claimants. In some cases, if you are talking about a transition in types of government, like Napoleon taking over republican France after the Revolution, some additional work is required to give the kingship (or in Napoleon's case emperorship) itself legitimacy. Successful resolution of domestic problems and successful foreign conquest are the traditional routes.