Stalin never officially named a successor, though Malenkov was likely the person whom Stalin "most" wanted to take over after his death. So what you had in the aftermath of Stalin's death was a power struggle between Stalin's inner circle to determine what kind of leadership would emerge. Initially, there was fear and hatred by nearly everyone in the circle against the person of Lavrentiy Beria: Stalin's secret police chief. Khrushchev and the rest of "team Stalin" managed to arrest Beria and executed him in December of 1953.

In theory, the Soviet Union was under a "collective leadership" during this period, much like what the Soviet Union had in the 1920s. No one person was suppose to hold all the power as Stalin did. Hence, Khrushchev was made first secretary of the party, while Malenkov succeeded Stalin's title as premier. Stalin did not actually officially hold the title of general secretary (he actually had the post abolished) at the time of death and thus technically Malenkov was Stalins' successor.

Khrushchev however, was able to consolidate his power and attacked Malenkov, accusing him of collaboration with Beria while he was alive and forced him to resign his post as premier. Though he retained other, lesser but still important posts.

In 1957 Malenkov, along with several other surviving members of Stalin's inner circle, attempted a coup against Khrushchev as they became angered by his reforms and increasing monopolization of power. They attempted to enact what amounted to a sudden "vote of non confidence" against him in the Soviet politburo presidium to remove him. The coup failed however, when Khrushchev rallied his allies and in particular the support of Marshal Gerogy Zhukov. Zhukov made it very clear that he had the armed forces behind him and that he would crush the coup with force if necessary.

Malenkov and his allies were therefore forced to back down. They lost all their high level political posts and forced into exile. They were however, not executed or jailed, Malenkov was made the manager of a hydroelectric plant in Kazakhstan and died in relative obscurity.

Therefore, while Malenkov could be said to have succeed Stalin in his post as premier, his rapid defeat by Khrushchev in the power struggle meant that he never gained the amount of power that would be expected of a Soviet leader. Hence why history considers Khrushchev as Stalin's de facto successor.

Source: Stalin and the court of the red tsar by Simon Sebag Montefiore

On Stalin's Team: The Years of Living Dangerously in Soviet Politics by Sheila Fitzpatrick