The How to New York articles from news site THE CITY are helpful guides to how New York City works, including navigating city and state laws and bureaucracies. Topics include Who to Yell At: A New Yorker’s Guide on Who to Call to Get Local Problems Fixed and How to Join a Community Board, and What to Know Before You Apply.
The New York City government is mayor-council system:
The Laws of the City of New York are comprised of:
New York City's official Organizational Chart (click to enlarge).

Before diving into budget documents, review the NYC Independent Budget Office's NYC Budget Guides for very helpful overviews of how the budget is made and where the money goes!
There is no single digital repository of city government publications. Below are common sources for publications, both current and historic: