While it may be somewhat overstating things to say that the Federalist Party met its demise in the late 1790s, it surely did decline. When political parties decline, it is hard to identify exactly what role specific people played in the decline. My own feeling is that John Adams did not do much to cause the decline of the Federalists even though he was president at the time.
The main reason for the decline of the Federalists was the simple fact that they were not as popular as the Jeffersonians. The Federalists’ domestic and foreign policies were both relatively unpopular among the people of the United States. It was for this reason that the party declined so badly in the late 1790s.
The Federalists as a group were really never all that popular in the United States. They (or a group that resembled them strongly) held power for the first three presidencies under the Constitution largely because of how popular George Washington was. Once he stepped down, there was less to make Americans feel any allegiance to the party.
The Federalists were, for the most part, a party of the elite. Their domestic policies, laid out largely by Alexander Hamilton, appealed to the economic and social elites of the country much more than they appealed to the common person. The unpopularity of many of Hamilton’s policies (such as taxes to fund a national debt) caused the Federalist Party to decline.
The Federalists also diverged from much of the country in their support for England over France. After the French Revolution, these two countries had come into conflict. Federalists supported the English for economic reasons and because they liked England’s constitutional monarchy much more than they liked the seemingly untrammeled democracy of France. This, too, made many Americans unhappy.
Adams was not a very partisan Federalist. He was much more of a moderate. He pursued policies (such as trying to make peace with France, that angered many Federalists. Therefore, I would argue that the decline of his party was not really his fault. The Federalist declined simply because their policies were not popular.