In the absence of George Washington, the main contenders to serve as first President under the new Constitution were John Adams and Thomas Jefferson. Alexander Hamilton wanted the presidency for himself, but was too unpopular to obtain much support. In a calculating move, he threw his support behind Adams, who won a close election of 1789.
Once Adams was in office, Hamilton proceeded to discretely undermine the new President's authority, through rumormongering and various legislative machinations in the Congress. Adams lost the subsequent election to Thomas Jefferson and his running mate, Aaron Burr. Due to the timing of Jefferson's election and service in office (1793-1801--becoming the first President to serve two four-year terms), Jefferson never had the opportunity to execute the Louisiana Purchase.
Hamilton himself because the Federalist standard-bearer in the election of 1800. Winning the presidency over Aaron Burr, Hamilton proceded to consolidate presidential power, using patronage to stack his cabinet with those he deemed would be loyal to Hamilton personally.
The effects of the Hamilton presidency were both acute and long-lasting: After the election of 1800, the presidency became the pre-eminent branch of federal authority. Madison vs. Marbury never took place, thus the concept of the judiciary ruling on the "constitutionality" of laws never developed. Additionally, although Congress retained the power to enact legislation, due to Federalist majorities in both houses of Congress during Hamilton's term, it became ordinary for Congress to do the bidding of the executive branch, and Congress never fully developed itself to be the check on Executive power that it became IOTL.