Leo Bokh studied art history, archaeology, history, German and musicology in Vienna and Graz. While still a student he joined the Cartellverband. After graduation he worked as a writer and art critic, for the Grazer Volksblatt and Deutsche Allgemeine Zeitung Berlin, among others. In 1934 he became a consultant in the culture department of the RAVAG state broadcasting company. From 1937 until the annexation in 1938, he was also a member of the local council as representative for the independent professions in Graz. Having previously criticized the National Socialists, his attitude changed when the Nazis came to power and around 1940 he even became a member of the NSDAP. In 1941 he was an assistant to Karl Garzarolli-Thurnlackh in the Landesbildergalerie (Styrian Painting Gallery) and Alte Galerie (old masters' gallery) at the Joanneum in Graz. Just a year later, the Joanneum arranged an exemption from military service for him because of the additional work required on account of the many new acquisitions at the time (dissolution of monasteries, seized collections, etc.) and the division of the Landesbildergalerie into an Alte Galerie and a Neue Galerie. Apart from Garzarolli-Thurnlackh, Bokh was the only academic in the Alte Galerie and was not therefore required to serve in the army. He managed the loan register and air raid depots, which were outside Graz.

Bokh specialized in Baroque art and after the war he enlarged the collection with numerous acquisitions, including items obtained on the art market. After 1945 he was classified as a lesser offender and in 1946 succeeded Garzarolli-Thurnlackh as director of the Alte Galerie, a position he held until 1956. He was mainly involved in reorganizing the Alte Galerie, but also in the restitution of works expropriated during the Nazi era, including the assets of Oscar Bondy, Valerie Eisler, Rudolf Gutmann, Albert Pollak, and Alphonse and Louis Rothschild. He was suspended from duty with a reduced salary in 1956 following two disciplinary proceedings on suspicion of embezzlement and irregularities in the inventories and retired six years later.