How did Joseph Stalin become the leader of the USSR?

While Lenin was disabled from strokes, Joseph Stalin, Grigori Zinoviev and Lev Kamenev ran the country through the structure of the Communist Party. This continued after Lenin died in 1924. Stalin was General Secretary of the Communist Party and began placing people loyal to him in important posts in government and on the Central Committee of the Party. He eliminated Leon Trotsky, Lenin's choice for his successor), had him expelled from the Party then from the country.

Over the next few years, Stalin isolated Zinoviev and Kamenev and had them expelled and eventually executed. By 1929, he was the only one left with any power in the government although he was still subject to being removed by vote of the Politburo of the Central Committee. By about 1932, he had enough of his men on the Politburo that he could do almost as he pleased without worrying about being removed. That is when he became the dictator that he is known for.