The primary reason is that Hamilton did not believe he could manipulate Adams to the extent he could influence Thomas Pinckney, the other Federalist candidate. Adams had opposed war with France, which Hamilton had heartily supported, so this may also have been a factor. The best explanation is that Hamilton wished to be the power behind the throne, and impose his own vision of America on the nation. He had made too many enemies during the Washington Administration to be elected in his own right; so he hoped to remain an unofficial power broker. Adams was too independent for Hamilton's taste. There apparently was some bad blood between the two already: Adams once described Hamilton as "the bastard brat of a Scotch peddler."
Interestingly Thomas Jefferson probably would have defeated Adams for President in 1796; but the French ambassador had publicly supported him, which backfired. Hamilton's plan to get votes for Pinckney also backfired, and the end result was Adams became President and Jefferson Vice President.
The major reason that Alexander Hamilton had for opposing John Adams' bid for the presidency in 1796 was the fact that Hamilton himself wanted to have more power.
During George Washington's time as president, Hamilton had been one of Washington's most important advisors. Hamilton was the leading figure among the Federalists in Washington's administration (Jefferson was the head of the Democratic-Republican faction under Washington). Hamilton would, therefore, have been the most obvious choice to succeed Washington. However, he was a polarizing figure and had made many people angry.
Hamilton still wanted power, though. He felt that Thomas Pinckney would be a better choice than Adams. This was because he felt that he could exert more control over Pinckney. Hamilton felt that Adams was too strong-minded and would be harder to control than Pinckney.
Actually the main reason that Hamilton did not back Adam's as president is that he did not believe he had the capacity to govern well.
The whole "Pinckney' as President thing was simply a way to further prevent Jefferson from attaining the office. . . coupled with the fact that Pinckney was a "southern" Federalist which would have a good effect IF he would somehow be elected. Adams simply was NOT a card carying Federalist either ... so clearly Hamilton had his misgivings. There is no evidence that Hamilton thought that he could "control" Pinckney ... or desired to at all. Adams was stupid enough to retain all of Washington's cabinet... who all greatly valued Hamilton's advice... and asked for it at every turn. The Alien and Sedition acts are examples of Adam's thin skin and inability to govern well. Hamilton advised against these acts. During Washington's Presidency Hamilton angered "some" people... due to the fact that he was usually RIGHT...and Washington backed him at nearly every turn.