How can I marry my boyfriend after he overstayed a visa?

Generally when a foreign national enters the U.S. lawfully and with inspection (as appears to be the case for your fiance in 2015) and then becomes married to a U.S. citizen, the couple can succeed in the application process by which the foreign national will "adjust status" to become a Lawful Permanent Resident (to get a "Green Card"). This is true regardless of whether the foreign national spouse may have overstayed his initial visa. As a Permanent Resident, he would be entitled to remain in, be employed in and travel abroad and return to the U.S. without being subject to deportation/removal. Generally within 33 months following success with adjustment of status, the foreign national spouse even could become eligible to apply to become a naturalized U.S. citizen. Of course, there are many other details that determine eligibility for adjustment of status. There really is no substitute for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.