How did the American public react to the Burr-Hamilton duel in 1804?

The outcry over the death of Hamilton was considerable, and it was expressed in a few different ways, of which I'll focus on several.

It is important to keep in mind that dueling is a complicated thing. It is probably safe to say that even in societies that practiced it fervently, it was decried by the simple majority, and rather its survival and important was due specifically to approval within elite circles - male ones, of course - where the practice was continually reinforced. So throughout its practice, there are often voices decrying it, even prominent ones, but it was very rare that they could break through and have any real impact (see here for more on this). Dueling's fortunes were thus tied usually to broad shifts in society over time. In the New England states by this time, it was already mostly a bygone practice. In New York, it still lived on in elite political circles. Only in the South would it really have a long life ahead of it still.

If we look at the northeast, there was certainly anguish over his death, but one of the most prominent sets of voices in the wake of this were religious leaders who say in his death a larger object lesson about virtue and life. Dueling, again, was already declining, and unpopular with many people, but this was a very prominient, and shocking event, and anti-dueling crusaders saw their opportunity. As such, a number of pastors preached sermons on the evils of dueling which were then printed for publication and wider distribution. One of the most famous of these was by James Abercrombie, which he delivered less than a month after the duel in Philadelphia. It is several dozen pages, which you can find here but I'd highlight to particularly notable segments:

Various indeed are the modes, in which the king of terrors exercises his power over the human race ; the most lamentable and distressing of which is that, whereby he renders men the agents of their own dissolution ; either by the perpetration of wilful Suicide^ or by the equally atrocious act of Duelling* — a practice which, notwithstanding the explicit and positive prohibition of divine and human laws, frequently prevails in the most civilized nations, and even amongst those who profess, and call themselves Christians. — A practice so deeply rooted, and so generally acquiesced in, so interwoven with the respect and esteem of our fellow men, as often not to be resisted, but under pain of forfeiting these, and the ability of future usefulness in life. A practice, moreover, which I am sorry to observe, is rapidly gaining ground, and its advocates daily increasing amongst us ; though it is known to be an act, replete with danger and distress, ferocious in its nature, savage in its operation, and impiously antichristian in its principle. [....]

[I]f neither the dictates of reason, the persuasions of religion, nor the absurdity of those impious principles of false honour, which involve men in the guilt both of voluntary and intended Suicide and Murder (unquestionably incurred by the duellist,) — If none of these powerful motives can restrain him, let him listen to the voice of humanity — let him consider the duty which he owes to society ; and the unmerited misery into which he may suddenly plunge the innocent and virtuous relatives and dependants of his unfortunate antagonist. Having satiated his Revenge for a supposed injury; his Jealousy of his superior success ; or his Envy of his unrivalled and acknowledged talents ; — let him view him as the victim of his resentment, prostrate on the earth, weltering in his blood, and writhing under the excruciating agony of a mortal wound. Let him follow him from " the field of blood," to the chamber of death — see him in the last agonizing moments of dissolution, surrounded by his friends — his distracted wife bending over his almost lifeless frame — and, perhaps, a group of helpless children swelling the tide of woe with the most heart-rending sobs and lamentations.

Abercrombie was joined in this exercise, and these sentiments, by pastors around the northeast, with similar sermons to be found in New York and New England. Dueling had already been seeing a decline in societal approval in Northern states, and Hamilton's death helped push that view onwards. Dueling as an act of suicide was a popular motif for anti-dueling activists, and with Hamilton especially this was a theme they latched on to. A different sermon, that by Eliphalet Nott delivered in Albany, provides a good distilling of this when he talks of Hamilton's crime being the exposure of his life, and Bishop Benjamin Moore, President of Columbia College, upon hearing of the event cried out "Let those who are disposed to justify the practice of dueling, be induced, by this simple narrative, to view with abhorrence that custom."

Nott's commentary there of course is interesting to contrast with Hamilton's own words which he wrote out prior to the duel, laying out his conflicted mind and stating how he wished he didn't have to duel but he nevertheless felt it necessary as not to do so would be shameful:

I answer that my relative situation, as well in public as private appeals, inforcing all the considerations which constitute what men of the world denominate honor, impressed on me (as I thought) a peculiar necessity not to decline the call. The ability to be in future useful whether in resisting mischief or effecting good, in those crises of our public affairs, which seem likely to happen, would probably be inseparable from a conformity with public prejudice in this particular.

It is a consistent theme, where the culture of honor demanded risk of life and limb to order to avoid shame; and contrasted with this alternative worldview where it was guilt, and internal conscience that guided actions, not an attempt to avoid public shaming. There is some irony, then, in the fact that Hamilton's appeal to the former helped solidify cultural shifts to the latter. To be sure he doesn't deserve sole credit, his death coming in the middle of such changes, and merely providing a lightning rod to use as illustration, rather than itself an impetus.

However, while a large focus of the outcry was on the act of dueling itself, and a great amount of ink was spilled and voices raised on how Hamilton's choice to duel at all was a poor one which he ought not have done, it was generally coached in terms of lamentation, rather than invective. That was mostly reserved for Burr, which was perhaps the greatest irony of the entire affair!

In issuing his challenge, Burr believed he was going through an established ritual of honor. His political career was feeling stalled, and he believed Hamilton to be at fault. He had written Hamilton to demand a retraction which he could then make public, and which would help revitalize his political fortunes; or else if no retraction was forthcoming, he would be able to demonstrate his manly virtue through the act of the duel, which likewise would have the same effect. Had Hamilton lived it is hard to say whether he would have succeeded there, but Burr's early biography Biddle certainly believed a non-fatal result would have assured Burr the governorship of the state. Dueling was not infrequent in New York politics over the decade leading up to the duel (theirs was only one of 19 fought at Weehawken, almost all involving New Yorkers), and other duelists had come out well enough from such acts. But Hamilton's death of course changed things.