Also see the Alexander Hamilton Collection: The Story of the Revolution & Founding.
The Constitution gave Congress the right to determine the process by which foreign-born residents could obtain citizenship, and a 1790 Act of the First Congress laid out the process. This 1795 revision required all persons who wished to become naturalized citizens to go to a court to declare their intention at least three years prior to formal application. They would have to take an oath of allegiance, be a person of good moral character, agree to support the Constitution, and renounce any former sovereign and hereditary titles.
“any alien, being a free white person, may be admitted to become a citizen of the United States, or any of them, on the following conditions, and not otherwise....”
By limiting naturalization to “free white” persons, the early acts effectively prevented any people of color or indentured servants from gaining citizenship. Over the next century and a half, these restrictions were at first reinforced (for instance in the notorious Naturalization Act of 1798, part of the Alien and Sedition Acts, which extended the required residency period to fourteen years), but then eventually eliminated by subsequent revisions.