Sir Francis Bacon
As a nonfiction work, The New Organon does not have characters in the traditional sense. It does not, however, come from a void, and the character of the author must be taken into account to fully understand the book. Francis Bacon was both an intellectual and a politician, and his cunning countercultural tendencies were on full display in both areas of his life - in politics, he was convicted of corruption, while in the intellectual sphere he rebelled against the traditional wisdom of the ancients and revolutionized conceptions of the field of logic and reasoning. His reactions against Aristotle, while rational, might therefore be understood in context of Bacon's character, slightly altering the reading of this work.
Additionally, this book was written after his fall from power, which is something to take into account when reading as well.
Aristotle
Much of the New Organon's substance concerns the work of Aristotle, the ancient Greek philosopher whose ideas held dominance in intellectual spheres during Bacon's time. Bacon violently disagreed with Aristotle's methods of reasoning, specifically the blind acceptance of some truths a priori and without experiential evidence, and he therefore wrote much of this book as a counter to Aristotle. Even the title is an irreverent reference to Aristotle's work: the Organon is the collection of Aristotle's volumes on logic, and by titling his work The New Organon, Bacon is boldly making a statement that he has bettered Aristotle.
James I
Bacon dedicates the work to James I, the reigning monarch at the time, entreating him to implement these ideas into his rule. By doing this, Bacon is also flattering the king, whose favor might be enough to reinstate his reputation. James I is therefore another important character, knowledge of whom provides crucial background knowledge for understanding the context of The New Organon.