What was the United States' relationship with Napoleon like?

Based on some brief reading, it seems the relationship between Napoleon and the U.S were strained, to say the least, by the desire of the United States to remain mostly neutral in the worse relationship between France and Britain. It should be noted: Napoleonic France was more occupied with Britain, and the salt of a recent revolution was still fresh between the U.S and Britain. It is difficult to talk about the relationship between the U.S and Napoleonic France without involving Britain.

The story of U.S and Napoleonic France seems to start back in 1798 with an undeclared Quasi-War between France and the United States, negations of which to bring it to an end were in full swing as Napoleon rose to power. Napoleon, as many French at the time, very much distrusted the British, but did sign the Peace of Amiens in 1802, ending a decade of war. At this time, Jefferson was president, and expressed concern over the leadership of Napoleon, but was more concerned with Britain. The Louisiana Territory held by the French was intended for use as a holding by France, having been re-obtained from Spain, but the Haitian Revolution meant the Louisiana Territory was no longer a good territory, leading the the Louisiana Purchase in 1803, and a declaration of what would be a decade-long war by the British against France. This conflict, and the Impressment Acts used to seize ships and sailors for use in war, led to a deterioration in relationships between the U.S and Britain.

In 1806, both Britain and France passed treaties forbidding trade with one another, an act the U.S mostly ignored, preferring to remain neutral in the conflict. However, following the June 22, 1807 bombarding and boarding of a U.S ship by the British, President Jefferson signed the unpopular embargo on all foreign trade in an attempt to harm the economy of Britain, an act that proved next to impossible to enforce. The Non-Intercourse Act followed in 1809, forbidding trade only with Britain and France. It was also unpopular and difficult to enforce.

It should be noted: Britain and France were both still at war, and both wanted trade with the United States. Jefferson opted not to run for president in 1808, and Madison won the election. Now: Madison was in a bit of a pickle: the Non-Intercourse act was ineffective, but it was intended mostly as a stab against the British. To repeal it would be to effectively submit to the British terms of trade in international waters. In 1810, Madison signed Nathaniel Macon’s Bill No. 2, offering a return of trade to both France and Britain, in exchange for ceasing their practice of seizing U.S merchant vessels. Privately skeptical of French intention, they were nonetheless the first to offer concessions, and Madison renewed trade with France in 1811. This proved massively detrimental to U.S.-British relations. It is well known the War of 1812 was fought against the British, but not as well known that war was also considered against the French for many of the same reasons. In 1812, Madison declared war on Britain, which would last to 1815.

Overall, the years from 1798 to 1815 were what would be expected of a minor nation trying to remain neutral in a conflict between two major world powers. The U.S. was mostly ineffective at brokering any kind of peace between the two, and showed the United States were still weaker than Britain. The U.S and Napoleonic France did not particularly get along in the early years of Napoleon's rule, with trade and diplomatic relationships mostly severed.

Sources: A People's History of the United States, Empire of Liberty: A History of the Early Republic, 1789-1815 (Oxford History of the United States). I don't have any particular history qualification, and may have gotten some details wrong. Please correct me if I did.