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For the entirety of act one Jefferson spent his time in France negotiating treaties and making powerful allies, most notably with that of Marquis de Lafayette, who was friends with Hamilton prior to returning home. His reasons for going to France are presumably due to his political duties but also possibly because he needed a distraction and form of escapism from his wife passing away (as was the case in the actual historical figure's life).
After returning from France to Virginia Jefferson found a letter from the president summoning him to a cabinet meeting and declaring him as the secretary of the state. He enthusiastically headed to New York only to be met with his best friend James Madison telling him about Alexander Hamilton and his new financial plan, and after a debate with the latter he and Madison mocked Hamilton on the fact that he didn't have the votes to put his financial plan into play.
Later he and Madison decided to hear what Hamilton had to say in a separate, private meeting were they eventually came to a compromise and moved on. Shortly after this however another debate occurred on whether or not America should assist France during their revolution, with Jefferson being in favor of doing so due to not only his close connections with the country but also because he finds it a dishonor to America's treaty with France previously to refuse assistance. Unfortunately Hamilton and George Washington shoot down his claims and he confronts Hamilton on this, reminding him of their mutual close friend Marquis de Lafayette, Hamilton once again shoots down Jefferson however, prompting him to remind Hamilton that he's only in a secure position of power thanks to Washington's trust in him. Alongside Aaron Burr and James Madison he laments over how it must be to be on the same side as Washington before having an epiphany and realizing a way to get power over Hamilton.
He resigns from his position as secretary of the state and conspires with Madison and Burr to dig up dirt on Hamilton, eventually finding irregularities in his financing and concluding that he was performing illegal activities under the name James Reynolds. After they confront Hamilton about this however he reveals the truth to them, that he had an affair with one Maria Reynolds and was blackmailed by her husband into supplying him with financial support, in spite of knowingly consenting to the affair and even allowing them to continue it. Although horrified to discover this Jefferson and Madison both promise to not spread the news, Burr on the other hand instead tells Hamilton that "rumors can only grow".
When Hamilton himself decides to release a pamphlet detailing his affair with Reynolds, Jefferson and Madison took pleasure in spreading it around, satisfied that now they didn't have to worry about Hamilton getting in the way of Jefferson's run for presidency due to his reputation being ruined. Shortly thereafter Alexander Hamtilon's son Philip gets killed in a duel, and Jefferson shows him mild pity after Alexander and his strained wife Eliza decide to move uptown in grief.
Jefferson is then in a three way battle against Aaron Burr, John Adams and himself in a new run for presidency. Eventually the decision is put to a tie between Jefferson and Burr with Hamilton having to give the deciding vote. To Jefferson's pleasant surprise Hamilton picked him, leading him to become the third US president. Aaron Burr although hurt congratulates Jefferson and told him he looked forward to their partnership, Jefferson scoffed at Burr however, instead appointing Madison as his vice president due to Burr being against him for such a long portion of their careers.
Shortly after this a humiliated Burr challenges Hamilton to a duel and manages to win a panic, killing Hamilton but instantly regretting his decision. Overtime Jefferson came to feel remorse for Hamilton's death and even admitted his financial plan was a work of genius, so much so that Jefferson in spite of his attempts couldn't undo it.
After stepping down as president he was succeeded by Madison and eventually he passed away ironically enough on the 4th of July in 1826 of presumably natural causes due to the fact that he would have been 83 years old at the time.
Jefferson was a loud, flamboyant, suave, intelligent and jovial man, with a great sense of charisma, but was also quite eccentric, outspoken, passionate and hypocritical, due to his preaching of the ideals of freedom in spite of being a slave owner. He was also fairly elitist, looking down on Hamilton for being an immigrant and coming from new money, he also took great pride in Hamilton releasing the Reynold's pamphlet and essentially destroying his own career, taking great delight in spreading the pamphlet knowing that he had one less political rival to worry about.
In spite of that however Jefferson did some good qualities, for one he was a man of his word, and genuinely did promise not to tell anyone about Hamilton's affair (although this didn't stop him from spreading the knowledge after Hamilton himself did), he genuinely loved and respected the French and was passionate to help them in their hour of need, and as a whole was a great admirer for all of America' s ideals. He was very close to James Madison, a fellow politician, even showing concern for his worsening cough over the course of the show. Finally he did respect Hamilton as a person in spite of their differences and felt genuine remorse after his death. He also showed similar sympathy when Hamilton lost his child Philip, as he too knew the pain of such an experience.
Jefferson was a tall, slim, handsome, dark skinned man, with a thick mass of dark brown curly hair, springing out of his head, and a neatly trimmed beard. He wore a crushed velvet violet suit paired alongside white tights, black shoes and at first wore a satin lilac longcoat that he later swapped out for a velvet violet one to match his suit. He also usually carried a black cane on occasion.
In comparison to his real life counterpart, Thomas Jefferson had a wildly different personality in the show than he did in real life. While both versions did play a similar role in America's history and Alexander Hamilton and Aaron Burr's political careers, while both versions share their eccentricities, wealth and ties to slavery Thomas Jefferson in real life was rather shy, introverted and quiet whereas in the musical he was a jovial, flamboyant and charismatic man.
It is understandable why Jefferson sympathized with Hamilton much more than before after the death of his son Philip, considering that historically four of Jefferson's children had already died young by the time of Philip Hamilton's death.