Criminal justice in USSR in the post war period was performed by "People's Courts" (they existed in 1937-1989). They consisted of judges and "people's jury". The judges were elected by popular vote for the term of 5 years. The jurors were elected for 2 years by meetings of "labor collectives" (factories an other enterprises). All elections in Soviet Union were strictly controlled by the Communist party, so usually 99.9% voted for a single candidate proposed by CPSU.
There was no such notion as "independent court", even in theory. The system of justice was "a branch of the government". In addition to this, aspects of life, all institutions in USSR, including the government and justice were strictly controlled by CPSU, and this was written into the constitution.
Superficially, the process looked as in democratic courts: there was an accuser and a defender, the judge presided and the jurors made decision by vote. In some cases the jurors were not used and the judge made the decision him/her self.
There was also a variety of other courts, "Camarade's courts", tribunals, etc.
But the usual criminal cases were handled by the People's courts.
In the period immediately after WWII, enormous number of people were imprisoned in camps or exiled without any court hearings, for alleged "collaboration with enemy", or waging war against Soviet power. (Whole nations were deported, and
other nations decimated). In most cases, the armed resistance against Soviet power was qualified simply as "banditism", but it was not handled by any courts.
Most of the camps population in the late 40s consisted of these people, though the ordinary crime rates were also very high.
Torture was not officially permitted after WWII, but in fact it was widely used.
Even in 1980s, beating by police was a routine "investigation procedure". Very few people dared to complain.
Death penalty was abolished in 1947, but in 1950 it was re-introduced for "spying and high treason". 10 years later people were sentenced to death for
"illegal currency transactions". So the state did not respect its own laws. But in any case, death penalty was very rare in comparison with 1930s.
See, for example, http://lawbook.online/gosudarstva-prava/organyi-yustitsii-sud-prokuratura-29805.html
Sorry, in Russian.