When thinking about all the many meanings of the simple word bill think of the bill you get in a restaurant. A piece of paper that someone pushes toward you, right?
In the legislature, laws draft form are called bills––pieces of paper being passed around. Dollar bills and advertising bills get passed around too. A bird will push the bill of their beak into things too, and the bill of your baseball hat is the part that sticks out in front of the rest of you. When an actor demands top billing, they want to come first on the billboard marquis in front of the theater (and everywhere else).