How did The Federalist Papers help to define the national government?
The Federalist Papers helped to define the national government in at least two ways.
First, they helped to explicate what the roles of the various parts of the government should be. They set out the views of the Framers (or at least those Framers who were Federalists) about things like the proper role of the judicial branch (Federalist 78) or the reasons for having a one-man executive (Federalist 70).
Second, they laid out the argument for consolidating power in the national government as opposed to leaving it with the states. This was a huge change in a country whose revolution had been based on a distrust of centralized power. In Federalist 10, Hamilton lays out the reasons why a large republic is better than a small one. This really helped to define the national government as a protector of liberty.