The genesis and purpose of Jefferson's composition of his Gospel harmony is set out pretty clearly in an 1804 letter that Jefferson wrote to noted unitarian thinker Joseph Priestly. In the letter, Jefferson provides some advice to Priestly regarding Priestly's own plan to write out a study of the moral teachings of Jesus. In giving his advice, Jefferson reveals the early stages of his own investigation in that topic:
I think you cannot avoid giving, as preliminary to the comparison, a digest of his moral doctrines, extracted in his own words fro the Evangelists, and leaving out everything relative to his personal history and character. It would be shore and precious. With a view to do this for my own satisfaction, I had sent to Philadelphia to get tow testaments (Greek) of the same edition, and two English, with a design to cut out the morsels of morality and paste them on the leaves of a book, in the manner you describe as having been pursued in forming your Harmony. But I shall now get the thing done by better hands.Letter to Joseph Priestly, January 29, 1804, reprinted in In God We Trust: The Religious Beliefs and Ideas of the American Founding Fathers, ed. by Norman Cousins (Harper & Brothers: 1958), pg. 171.
The letter from Jefferson to Priestly provides key insight about the context surrounding the Jefferson Bible. First, Jefferson was not engaging in a unique activity. He wasn't the only one engaging in the production of the Gospel harmony designed to spread a vision of Christianity that was grounded on the moral teachings of Jesus, rather than in the New Testament's teachings about Jesus. Jefferson notes at the end of the passage quoted above some relief at the idea that Priestly is working on a harmony because of Priestly's greater skills at producing such a study.
Second, Jefferson's advice indicates the key to this method: focusing on the words of Jesus rather than the explanation of those words provided by the Evangelists and the other writings in the New Testament. Jefferson's approach fused an antiquarian approach to the Gospels -- trying to get back to the earliest strata of the teachings of Jesus -- with a belief that the Gospel accounts as we have them were an accurate source of those words. Hence, Jefferson sought not only to take the words of Jesus from the New Testament as translated into English, but he sought to take the words from the Greek New Testament as well. It turns out that Jefferson was incorrect about the words of Jesus in the Gospels being the earliest strata of information we have about Jesus -- the epistles of St. Paul were actually written earlier than any of the Gospels as we currently have them -- but that's an error of application, not of method.
Third, at the time of the letter to Priestly, Jefferson sought to undertake his harmony for his "own satisfaction." Jefferson didn't seek to publish his study at this point, but was thinking of compiling his study for his own use. The nature of how he sought to carry out the composition of the harmony -- cutting out the relevant texts from his copies of the New Testament and pasting them into the "leaves of a book," indicates that he wasn't thinking of sending the text to a publisher. It was to be a private book indicating his own private thoughts, something quite understandable given the fact that Jefferson was a sitting president, an active politician, and one who was constantly dogged by allegations of infidelity regarding religion. For good reason he might seek not to widely publicize his views regarding the nature of Jesus of Nazareth.