You may enter in a BONA FIDE marriage (meaning not for the sole purposes of skirting the immigration law, but a genuine authentic marriage) and AOS in the United States, provided you meet all the statutory requirements like J-1 Two Year Waiver, absence of criminal history etc.
Speak with an immigration counsel in person before your next move.
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You must first become a permanent resident before you can become a citizen. You can apply for resident status ("a green card") through your boyfriend once you got married if he is a U.S. citizen.
Schedule a consultation for you and your boyfriend with an immigration attorney. Adjusting status is a complex process and it is not clear whether you have a home residency requirement, among other factors.
Make sure that you are not subject to the 212(e) known as the 2-year foreign residency requirement. If so, you will need an waiver before you can adjust your status to lawful permanent residency.
Myron R. Morales, Attorney
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As people noted you have to be careful the 2 year foreign residency requirement (see if the visa in your passport says you are or are not subject to 212(e). You may also have an issue if he ships out overseas. They have ways of dealing with that as well but you should use the assistance of an attorney of that is a possibility or if you're subject to 212(e).
My answering this question does not form an attorney-client relationship. Always retain a qualified attorney before taking any action.