We found 4 Reddit comments about Amsterdam: A History of the World's Most Liberal City. Here are the top ones, ranked by their Reddit score.
I'd say these two for starters:
Amsterdam: A History of the World's Most Liberal City - Russell Shorto
Amsterdam - Geert Mak
You may want to check out Russell Shorto's Amsterdam knowing him it will explain the goevernment of the city. I can't give you a sourced answer yet (this book is currently on it's way to my house), but what I remember from my school years government in the Dutch Republic was still the way it was in the Middle Ages. In the case of Amsterdam that would mean a council of free "poorters", citizens in English. These were elected by the free poorters from amongst themselves, of course there were some requirements to become a poorter, mainly financial and religious.
That council would chose a number of "burgermeesters", mayors in English and other magistrates. These would be the daily government. The council would also choose the cities representatives in the States of the Province, and ultimately the representatives in the States General, which governed the Republic. Amsterdam was of special importance, because on it's own it held the majority in the States of Holland, and since the States of Holland had the majority of votes in the States General, Amsterdam was very influential on the Foreign policies of the Republic. Please use this not as fact, this is what I remember from school some thirty years ago, but it may aid in your research.
Haven't read Mak yet but comparable I assume is Russell Shorto with Amsterdam: A History of the World's Most Liberal City