A Kentucky native, Schramm attended Western Kentucky University in Bowling Green, Kentucky and majored in English while also becoming intensely involved in theater. At the urging of his college acting teacher, he applied for and won a four-year graduate scholarship to Juilliard's newly formed drama school in New York.

During the summer of 1988, Schramm traveled to California to co-star with Rebecca De Mornay in "Born Yesterday" at the Pasadena Playhouse. This highly acclaimed and popular production brought Schramm to the attention of Hollywood, and he has worked steadily in television and film ever since.

Schramm has guest-starred on numerous television series, including Jake and the Fatman (1987), Wiseguy (1987), The Equalizer (1985), Miami Vice (1984) and Spenser: For Hire (1985). He also provided the voice for one of the characters in the Disney animated series Hercules (1998). In addition, he has had leading roles in The Time of Your Life (1976) and the musical Cradle Will Rock (1999), both for PBS.