How does each chapter from Founding Brothers by Joseph J. Ellis relate to his argument that the formation of America was shaped by the view of the key figures involved and their relationships with each other—whether they were unified in decisions or divided?

Ellis's approach is to focus upon the interactions among the major "players" in the early history of the US. Each chapter of Founding Brothers illustrates these personal connections and, paradoxically, stresses that disagreements among the founders, the conflicts that were played out in the three decades or so beginning in 1789, created the United States as we know it.
These conflicts were both personal and ideological, and Ellis depicts them on an intimate level that has generally been an untold part of history. The duel between Alexander Hamilton and Aaron Burr is a fairly well-known event, but the connection between the human conflict that caused it and the developing identity of the American people is less familiar to us. Similarly, in the next chapter Ellis discusses the Jefferson-Hamilton antagonism in relation to the wider issue of North vs. South and the questions of economic control coupled with the symbolic decision of where to locate the capital.
The chapter dealing with the decision of the founders to basically put off resolving the slavery issue again shows an aspect of personal interaction in explicit decision-making that most Americans are unaware of. Often the question is asked as to why these men did not have the foresight to outlaw slavery from the start. The facts become more puzzling on the surface when we realize that even those "founding brothers" who were practitioners of slavery knew that it was wrong. Ellis's account is a kind of paradigm of those instances in history overall when it's decided that one issue, however significant, must be subordinated to a wider plan that takes precedence, even at the expense of what is fair and moral.
In dealing with the Adams-Jefferson association, Ellis captures, in the personal enmity and then restored friendship of the two, what is perhaps the most central ideological conflict of the early US between Federalism and Republicanism. That the two men were eventually able to reconcile is symbolic in its way of the capacity the new country had to resolve the differences inherent in the great "experiment." Taken as a whole, all the episodes dramatized by Ellis are a metaphor of the basic idea that nationhood is a result of both conflict, often seemingly insoluble and solidarity. Each chapter of Founding Brothers exemplifies a principal truth Ellis states in his introduction: that the men and women who formed the new country, while they attempted to solve its problems in an unplanned, improvisatory manner, were also aware that they were part of a great drama that would be immortalized by posterity. And yet, these were people who, in the old world of Europe, would probably have never been able to rise to the level of carrying out actions that would change the course of history. Both their successes and failures created the country as we know it.

In the first chapter of Founding Brothers, Ellis explores the 1804 duel between Alexander Hamilton and Aaron Burr that ended in Hamilton’s death and Burr’s disgrace. Though they had both fought in the American Revolution and worked together as lawyers in New York City, their relationship ultimately deteriorated amid the political battles fought over how best to build the new nation’s government.
Chapter 2, “The Dinner,” covers the agreement between Hamilton and his nemesis, Thomas Jefferson, to move the national capital to the Potomac River (closer to Jefferson’s native Virginia) in exchange for Jefferson’s support of Hamilton’s plan for the federal government to assume states’ debts, which Jefferson and his ally James Madison had steadfastly opposed.
The title of chapter 3, “The Silence,” refers to the collective failure of all of the founding fathers to address the issue of slavery, instead leaving the explosive debate over the “peculiar institution” to tear apart the country along sectional lines less than a century later.
Chapter 4 examines President George Washington’s famous Farewell Address in 1796, in which he warned against the dangers of political factions (or parties) and foreign entanglements, among other threats he saw to democracy. Washington’s decision to step aside from the presidency opened the door to a heated battle between the Federalists (including Hamilton and John Adams) and the Democratic-Republicans (Jefferson, Madison and their supporters) over the future course of the young nation.
Chapters 5 and 6 focus on three important relationships: John Adams and his wife, Abigail, who famously urged him to “remember the ladies”; longtime allies and fellow Virginians Jefferson and Madison, who became the third and fourth presidents; and the renewed friendship between longtime rivals Adams and Jefferson once both were out of power. Their correspondence reveals both men reflecting on and debating their respective legacies and continued until the two men died within hours of each other on July 4, 1826, the fiftieth anniversary of the signing of the Declaration of Independence.