I read that the USSR had three branches of government, with the executive branch approving bills passed by the legislature, but also being compromised of the Presidium and the Council of ministers. Nominally speaking, the Presidium chair was the Head of State whole the Council chair was the Head of government. Yet the USSR had a communist party which was huge and complex, which had its own executive branch and legislature, headed by the General-Secretary. Reading about Soviet leaders, all except Lenin held the post of General Secretary on top of another important post. Who was in charge? Who checked whom? Was the communist party more powerful than the official government? Who would be considered the leader of the USSR?
Was the communist party more powerful than the official government?
In the Soviet Union the Communist Party and the Soviet state was, until its very end, almost synonymous with each other because the party held a monopoly over political power in the union. The Soviet government, designed under the principle of Lenin's democratic centralism, ensured that the Communist party would always control the state to the exclusion of other groups.
In reality the most of powerful post in the Soviet Union was that of general secretary, which made one the head of the party and hence the de facto leader of the Soviet Union. The top party official was therefore more powerful than the top state official. Every leader of the Soviet Union after Lenin held the position of general secretary.
The Soviet Union's nominal head of government was premier, and the nominal head of state chairman of the presidium. However, the leader of the Soviet Union did not always hold those posts. The position of head of state was mostly ceremonial.
Who checked whom?
Nominally speaking the general secretary could be checked by the Politburo and the central committee, who could vote to oust the secretary and replace him. In effect however, Soviet leaders were highly dependent on personal and patronage networks to ensure their stay in power.
An example would be the anti-party group who attempted a coup against Khrushchev and mustered enough votes in the Politburo to remove him. However, Khrushchev had the support of Marshal Zhukov who was vital in preventing the removal later on in the central committee. Later on however Brezhnev and his allies were able to remove Khrushchev after he had alienated enough of the party leadership through overly autocratic governance and the Cuban missile crisis. Institutional checks on Soviet leaders were weak, however the leader could usually be removed if enough of the top party leadership opposed him.