Bank-Mikkelsen studied theater history at Copenhagen University, and subsequently applied to Aarhus Theater's acting school. He graduated in 1969 and was there after connected to Aarhus Theater until 1978. During the 1980s, he could be seen in many of the Copenhagen theaters, including the Royal Theater, where he still performs to this day.
In 1975, Bank-Mikkelsen made his TV debut in the mini series Aladdin where he had the title role. Due to the depth of his voice, he has frequently been utilized by DR in readings of literature on the radio and as narrator on documentaries and fiction series, such as Bryggeren.
His voice acting credits include the Danish animated feature Valhalla (as Odin and Utgård Loki), The Polar Express (as Santa Claus), the Disney movies Aladdin (as Jafar), Hercules, and The Little Mermaid (as Flotsam and Jetsam), The Simpsons Movie (as Grandpa Abe SImpson), and many others. He is the official Danish voice of Eeyore in the Winnie the Pooh franchise.