Having recently read the biography of Alexander Hamilton, it occurred to me that there is a possibility that the requirement, written into the Constitution, that the President be a "natural born citizen" was inserted by Alexander Hamilton's enemies to prevent him from rising to the office as a hero of the revolution, despite his West Indies roots? I've seen little discussion in the Federalist papers regarding the intent of the founders.
No, because it wouldn't have had any effect. The relevant clause reads "No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution..." Hamilton would have been a citizen at the time the Constitution was adopted.
First off, it should be mentioned that the provision would not have excluded Hamilton at any rate, since it explicitly does not apply to anyone who was a US citizen at the time the Constitution was adopted. That, in and of itself, does not of course mean that his enemies weren't the ones who pushed for its inclusion, just that if they were then they did not do so with an eye towards excluding Hamilton (unless, of course, they were just really stupid).
With that said, it is worth noting that Hamilton himself had included such a requirement in a draft proposal he submitted to the Constitutional Convention. The committee that was responsible for the proposal that eventually did make it in to the adopted Constitution indeed included some people who disagreed with Hamilton's preference for an expansive, powerful central government (most notably George Mason and especially Elbridge Gerry) and so could perhaps be called "enemies," but even those disagreements did not necessarily expand to personal enmity (Gerry would later support Hamilton's central-banking plan, for example).
The reasons for the 'natural born citizen' clause are well-documented, and boil down to what you'd expect: to prevent 'ambitious foreigners' from seizing control of the US government.
Here's a really nice detailed breakdown of the background and reasons for the clause from the Harvard Law Review.