Klara Berkovich was born in Odessa, Ukraine (then part of the Soviet Union), the only child of Yefim Josefovich Gordion, a machinist, and Adele Raphaelovna Tesler, a teacher of Russian language and literature.
In 1934, Klara at age 7 began to study violin at Special Music School No. 1 with Viktor Karakes, a former student of Stolyarsky who in addition to teaching played in theatre orchestras.