I recently read Stephen Cohen's Bukharin and the Bolshevik Revolution: A Political Biography, 1888-1938, so I will attempt an answer but completely understand if the mods don't consider this an informed-enough response, or if someone with academic credentials can answer more comprehensively.

The early Soviet period (essentially the 1920s) was characterized by a somewhat informal structure that was largely centered around the Politburo (Political Bureau, a sort of leadership committee for the Bolshevik/Communist party), of which there was a changing roster of full and candidate members who would vote on various proposals that governed the party, and thereby the USSR itself. Each would often hold other official posts (e.g. Stalin was the General Secretary of the party itself, Grigory Zinoviev was head of the Communist International, Nikolai Bukharin was the editor of Pravda, the main Bolshevik newspaper). Lev Kamenev was a Politburo member and actually the first Head of State of Soviet Russia, but he was still only one of a series of Politburo members with an equal vote. Keep in mind that the early Soviet period was characterized by a lot of ad hoc ersatz leadership activities. The Russian Civil War led to what has been called "War Communism," or the suspension of formal clerical and governmental procedures in favor of doing whatever was necessary for the Red Army to triumph over the White (anti-communist) forces. This meant requisitioning grain and supplies instead of taxation, for example.

The Bolshevik (later Communist) party was not a disciplined, homogeneous organization throughout the 1920s, but was a coalition of people in trade unions, intellectuals, professionals, and others with various agendas and plans, and this period in the USSR was almost nothing like the later centralized Stalinist autocracy that would follow. Keep in mind that part of the reason Tsarist Russia was rocked by major revolutions in 1905 and 1917 was the backwardness and disorganization of a feudal monarchy unable to handle problems of the modern industrialized world. After the revolutions and the Civil War, the Communists essentially had to rebuild society, nearly from scratch, and they improvised and argued and debated, but at times didn't even have the power to enforce policies uniformly.

After the Red/Bolshevik victory in the Civil War, the Politburo was officially a majority-rule coalition of leaders, including Trotsky, Stalin, and Bukharin, but Lenin, as the main organizational hero of the October Revolution and the preeminent Bolshevik, was often deferred to. There were debates and disagreements, but party unity was paramount. The revolution and civil war had played out between the various revolutionary entities that dotted Tsarist Russian, including Mensheviks (who had split from the Bolsheviks decades prior), SRs (Socialist Revolutionaries) and various anarchist groups. Part of what led the revolution to victory was the cohesion of the Bolshevik (later Communist) party. Factionalism and division were considered to be harmful to the Soviet project going forward, and so while there were disagreements about policy, the Politburo and other Bolshevik leaders tended to outwardly support what's known as "The Party Line" and project a picture of unity.

That began to change when Lenin was incapacitated by a series and strokes, and then died in 1924. Quickly, factions formed, most infamously the Left Bolsheviks, headed by Politburo member Trotsky. While outwardly still projecting unity, within the halls of power there were intense disagreements. Part of the initial factionalism was over the policy of industrialization -- how fast it should proceed, how quickly farms and industry should collectivize, and ultimately how much state power and force should be used to coerce the peasants and populace of the young USSR off their land and into a modern industrial society. Trotsky and the Left were adamant that it should be swift, and moral considerations less important, and in this they were opposed (and defeated) by a coalition that included Bukharin and General Secretary Stalin.

With the Left defeated politically (although not imprisoned or purged -- this would come later), the final showdown would be between Bukharin and Stalin. Whereas Stalin had been a more moderate force against the Left, he took a more aggressive position vis-a-vis industrialization by the late 1920s, against Bukharin and what Stalin and his bloc now termed the Right. Why? Cohen argues that it was Stalin's nose for political power that drove him to adopt more-or-less the policy he drove out Trotsky for. War was brewing in Europe, and in order to quell a restive domestic population and prepare for eventual war, the future for power in the USSR seemed to be in industrialization. In contrast, Bukharin was a bit more of a humanist and wanted to "grow into socialism," rather than force it from the top. This is appealing in retrospect, but at the time the mainline of Marxist thought was that the confrontation with capitalist societies was inevitable, and so many (like Trotsky) saw the need to industrialize, build a strong proletariat, and prepare for this confrontation. Stalin adopted this line when it was politically beneficial for his accumulation of power. Additionally, It was still easy to appeal to the memory of the heroic revolution and stir up fears of a return to the feudal brutality of the Tsarist Empire. Bukharin, Stalin's last major opponent, was strongly against a "New Leviathan" of state power that would crush the peasants and workers under its heel.

So how exactly did Stalin accomplish this, and become preeminent leader of the USSR? Cohen argues that as General Secretary, he wielded enormous clerical and administrative power over the Bolshevik party itself. What does this mean? First, as the General Secretary, he drew up the agenda for major meetings, including the all-important Politburo conferences. So the very matters under discussion were chosen by him and his staff. But even further down the administrative chain, he was able to staff organs like the Leningrad or Moscow Soviets (party-controlled governmental councils that formed in the 1905 and 1917 revolutions) with loyalists. While there were still some people at this time in official government capacities who had been Mensheviks or SRs (they would be purged, imprisoned, and executed in the 1930s), the most ambitious and trusted functionaries were almost always Bolsheviks. By the time he was ready for an ideological showdown with Bukharin, he had a shadow government of administrators and employees ready to either swiftly take over those entities or at least fill them with a vocal minority that weakened the position of his adversaries -- like Bukharin. Before the Terror and the Purges, Stalin accomplished his consolidation of power with the stroke of a pen, Hence, he has been dubbed "The Gray Blur" for his ability to do paperwork and perform the typically boring or unexciting work that is so crucial to the management of modern organizations and states.

By the time of Bukharin's removal from the Politburo, Stalin's key allies were able to cleanly and legally rubber-stamp his policies and proposals. As his purges and show trials got under way later in the 1930s, there was no need to create new titles or roles for himself. He had sought power, and he held it, and remaining General Secretary was simply easier than trying to force a new model for government onto the very young, still-developing system. And it also allowed him the pretense that his rule was a legitimate continuation of the form it held when Lenin, beloved leader of the revolution, was still in alive.

I do recommend Cohen's book (Bukharin and the Bolshevik Revolution: A Political Biography, 1888-1938) if you are interested in this early period of Soviet history, what might have been, and Stalin's rise to power through the eyes of someone who could have been a much better leader -- or at least a much more human and empathetic one.