After Stalin's death in March of 1953, a troika formed between Georgy Malenkov, Lavrentiy Beria, and Vyacheslav Molotov. Matters of state became much less top-down unitary decision and proceeded to the quasi-democratic/oligarchic institutions of the Presidium and soviets. None of the three members of the troika held overwhelming sway and each showed some willingness to defer to the Presidium.

Beria was eliminated after his poor handling of the East German Uprising in June of 1953. He sent into divisions of Soviet tanks to violently put down a workers' revolt.

Khrushchev gradually amassed power by aligning key allies and lying just below the radar until he was ready to seize his chance. That chance came when Khrushchev delivered his "Secret Speech" to the members of the Presidium criticizing Stalinism and the Purges that characterized much of his era. When questions were sent forward after his speech, someone wrote "Why did you not question Stalin's policies before if you abhorred them so much?" Khrushchev then asked, "Who sent this question forward?" No one took responsibility. "Now you know why I did not speak up."

Khrushchev's new vision combined with a domestic popularity derived from his country bumpkin persona and common man way of carrying himself gave him the standing to pursue leadership.

Sources:

The Soviet Experiment by Ronald Grigor Suny

Molotov Remembers by Felix Chuev