Acevedo's parents were born in New York City and lived in the Bronx, where both their sons, Richard and Kirk Acevedo, were born and raised. Since childhood, Acevedo has shown an interest in acting; he performed before his family in improvised shows. Acevedo was a drama major in high school.
After graduating from LaGuardia High School of Music & Art and Performing Arts, Acevedo enrolled in the SUNY Purchase School of Acting (Purchase College). In the 1990s, Acevedo earned his Bachelor of Fine Arts degree and, as some of his classmates had landed parts shortly after auditioning for Oz, he was also motivated to try out, and subsequently landed the role of the half-crazed gang leader prisoner Miguel Alvarez.
