Borchard was born in Moscow. His parents were German. He spent his childhood in Saint Petersburg where he had music lessons. In 1920, after the Russian Revolution, he emigrated to Germany. Otto Klemperer made him his assistant at the Kroll Opera in Berlin.
He conducted the Berlin Philharmonic for the first time in January 1933. In 1935, he was banned by the Nazis for political reasons. He was not allowed to conduct, so he had to earn some money by teaching.
During World War II he was an activist who was working against the Nazis. After the war had ended he conducted the world famous Berlin Philharmonic in a concert which was very successful. One week later he was given the job of conductor of the orchestra instead of Wilhelm Furtwängler who was in exile in Switzerland. The part of Germany he was in was run by the Russians at this time. Because he had been anti-Nazi, and because he spoke Russian, he was favoured by the political leaders. In less than four months he gave 22 concerts with the BPO. In Berlin, as he was being driven home after a concert, he was shot dead by an American officer due to a misunderstanding at a military checkpoint.