Kopache was born in Manchester, New Hampshire, the son of Dorothy E. (née Sterling). He is known as Assistant Secretary of State Bob Slatterly in The West Wing and for various roles in the Star Trek franchise. He has also appeared in the stage productions of As You Like It as the Barnished Duke and in Antigone as Creon. He is one of few people who have appeared in all of the Star Trek television series, other than the original, and, along with Jeffrey Combs, J.G. Hertzler and Randy Oglesby, is one of only four actors to portray as many as seven characters.
His largest Star Trek role was as Kira Taban, the father of Kira Nerys (Nana Visitor) in Star Trek: Deep Space Nine. Kopache played Taban in "Ties of Blood and Water" and "Wrongs Darker than Death or Night".
Kopache had earlier played a Romulan in the Star Trek: The Next Generation episode "The Next Phase", a hologram in "Emergence" and a Starfleet officer in Star Trek Generations.
In 1994 he portrayed "Tu'Pari" in the Babylon 5 episode "The Parliament of Dreams".
Kopache makes a brief appearance in No Country For Old Men as a shoe salesman.
Kopache guest starred as Vic Feldspar, Craig Feldspar's lisping, gym owning, tight red jumpsuit wearing, Jack LaLanne-like father in the Malcolm in the Middle episode entitled, Living Will, from the show's sixth season.