How did Stalin eliminate the competition and gain power after Lenin's Death?

He removed all his political opponents from their positions and most of them were sent to Gulags.

Explanation:

After Vladimir Lenin's death in 1924, Stalin also intended to take control of the Soviet Union. At first, he removed many of the party from power and exiled to Europe and America, including Lav Trotsky, the man who was to take Lenin's place. The paranoia was spreading, and Stalin was pursuing his own terror, arresting people in the middle of the night and staging trials that looked more like performances. He blamed those he believed could be opposed to alliances with the capitalists and declared them "enemies of the people" and then executed. He then began conducting purges outside the party, accusing local officials of counter-revolutionary activities.