The answer to you question is definitely, "No."
When the Constitution was ratified, not every person was a "natural born citizen," so the language in Article II, Section 1 reflects that: "5: No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President." That clause means that any land-owning white Christian male at least 35 years old and residing in the U.S. for at least 14 years at the time of the Constitution's adoption was eligible to be President.
Hamilton was certainly eligible to be president, but his chances were the result of his own misbehavior. Once his affair with Maria Reynolds became public knowledge, his chances at being elected greatly diminished. And while it may have been possible for him to rehabilitate his public image in the long run, his life was cut short in his duel with Aaron Burr.