I'm a big fan of using historical empathy or historical perspective taking for teaching topics like this. If I were to teach the Federalist Papers right now, I'd start by studying the Federalists themselves. Look at the founders who were involved with writing about formation of a new government. How did people like Madison or Hamilton view the role of government and the balance between power and individual liberty in the new nation? How did the experiences of colonists under Great Britain influence how the Articles of Confederation were written and how did the limitations of the Articles lead to the federalist movement towards stronger central government. The Federalist Papers are not just documents, they are part of a story of how a government was born and evolved.
Of course, I wouldn't consider doing this without also looking at the Anti-Federalists who provided the other side of the debate. How did Patrick Henry, Thomas Jefferson or George Mason influence the evolution of American government? I'd introduce them when I introduced the Federalists. Present them as men who were interested in America's best interest but had very different ideas of how that should look. Make comparisons to politicians today who claim to know what's best for all of us. Then I'd look at the Federalist Papers and the Anti-federalist Papers as the written presentation of the arguments that people all over the country were having. Just like we do in politics today. These arguments were not just made by statesmen; they were being made in taverns, churches, meetinghouses, and homes all over the U.S.
Trying to bring a human face to the people behind the documents, but more importantly the ideas, from the famous to the not so famous, is how I'd try to get my students to see how politics has always been a debatable aspect of public life in the U.S. We often pretend as though the "Founding Fathers" believed this, or they believed that, as though they were a group of like-minded individuals who were somehow perfect creators of our republic. No, they were people, and just like any other person, they argued and had w3idely different ideas about what was best for the country.
I could see this ending with some form of performance-based activity in which the students were to not just recreate the debate over the Constitution, but to also evaluate the positions of the historical figures in relation to what we know about the Constitution today. Use the benefit of hindsight to contextualize the actions of the founders with the consequences of history. Speculate about the differences that may have occurred if the Constitution had been written differently. What if the Federalists had gotten their way (such as no Bill of Rights)? Would that have changed our history? Repeat that treatment to the Anti-Federalist position. Compare this with other great debates in recent American history that are also related to governmental power and liberty.
If there's one thing I dislike about the Federalist Papers it's that they are called "Papers". Who wants to learn about papers?