John Adams became president in 1799 when George Washington died. Because of his association with George Washington and therefore with the successful resolution of the French-American crisis, and because unlike in OTL there was no Alien & Sedition Acts John Adams wins the election. Hamilton is also supportive of John Adams, since there hasn't been enough time for a political rivalry to form. Thomas Jefferson becomes his vice president.
John Adams increased the size of the military, especially the navy. In 1803 he made the Louisiana Purchase. While the American navy had increased John Adams saw no reason to stop paying tribute to the Barbary states.
Rivalry With Vice President
Thomas Jefferson was harshly critical of John Adams often criticizing increasing tariffs, the increasing size of the military, as well as his failure to deal with the Barbary pirates.
The 12th amendment established that future elections would instead of having each elector cast 2 votes for president have each elector cast one vote for president and one for vice president.