He joined the Red Army in 1918. Kurochkin completed his cavalry courses in Petrograd in 1920, the year when he also joined the Bolshevik Party. He graduated from the Red Army High Cavalry school in 1923, the Frunze Military Academy in 1932 and the General Staff Academy in 1940.
In 1935 he was promoted to chief commander of a cavalry division. After the Soviet-Finnish war broke out, he took over as the commanding officer of 23rd Rifle Corps. From 1940 to 1941, he served as commander-in-chief of the 1st Army Group in Mongolia, commanding officer of the 17th Army in Mongolia, commander-in-chief of the Transbaikal Military District and commander-in-chief of the Orel Military District.
In July 1941 he was given command of the 20th Army which participated in the unsuccessful defense of Smolensk.
He briefly held the command of the 43rd Army in August 1941 until his promotion to commander-in-chief of the North-Western Front. He held that post until October 1942, when he was moved to command the Soviet 11th Army and then the 34th Army. He was again placed in command of the North-Western Front from July to November 1943. It was during this time that Kurochkin commanded the Toropets-Kholm Operation.
From December 1943 until February 1944, he was the first deputy commander-in-chief of the First Ukrainian Front under Marshal Konev and was remembered for his planning of the bloody Korsun–Shevchenkovsky Offensive.
From February to April 1944, he commanded the units of the Second Belorussian Front. His last assignment of the war would be commanding the 60th Army from April 1944 through May 1945 which took part in the Lvov-Sandomierz operation and fought in Central Europe. For all his merits and achievements, Kurochkin was awarded with the highest Soviet honor, Hero of the Soviet Union.
Immediately after the war, Kurochkin briefly headed the Kuban Military District until his appointment in 1946 as first deputy commander-in-chief of the Soviet Military Administration in Germany. After his tour in East Germany ended in 1947, he took over as an assistant commander-in-chief of the Far Eastern Military District.
From 1951 to 1968, Kurochkin served in military academies, first at the Military Academy of the General Staff and later as the Commandant of the Frunze Military Academy. In 1968, he became the chairman of the Supreme Command of the United Military Forces of the Warsaw Pact. Kurochkin finished his military career as a general inspector of the Ministry of Defense. He went on to become a deputy of the Supreme Soviet of the USSR and was awarded with the Order of Lenin in 1980. Kurichkin died in Moscow in 1989.