Leaders who got into the Politburo were expected to be a serving member of the Central Committee of the Communist Party, and were elected to the Politburo by the Central Committee. As intended, the Politburo members are the 'rapid response team' of the Communist Party, not a senior or elite body, and were nominated by the Central Committee to make swift executive action in times when a full Committee meeting or Party Congress was impractical.

So your first step is to become a well-respected member of the Central Committee. Becoming a Central Committee member in good standing is similarly a democratic process. Central Committee members are elected by the voting delegates of a Congress of the Communist Party and can come from any background whatsoever. Therefore, in a strict sense, any communist in the USSR could rise from a peasant to Premier if they had the approval of their peers at every step of the way.

Therefore, the power center of the party, the Central Committee, was meant to reflect the desires of the party more than an institutional power base. The Politburo, as the 'rapid response team' of the Central Committee, would also serve and reflect the desires of the party, rather than become a group of elites. This balance of powers actually functioned relatively well, with a lot of similarity to the early United States if you examine it, but it entirely collapsed once Stalin purged the government. But since you want to know about the Post-Stalin Politburo, you gotta deal with Stalin.

The Problem of Stalin (hoo boy)

Put a pin in Stalin.

During Lenin's lifetime, CC membership was decided by the voting delegates of each Communist Party Congress, held every three years. The freeness of this process is up for debate, but it did yield a number of opposition factions within the communist party, requiring Lenin to eventually ban factionalism (and a number of other leftist movements) as detrimental to CC business.

By the time of Lenin's death the CC had become pulled in two directions: first, by the ideological sway of Trotsky (who argued for democratic control of the party, the cynical estimation of which is that this was advantageous to him, but probably principled to some degree as well) and second, by the institutional pull of the expanding powers of the General Secretary of the CC, Stalin. Under the tenure of Stalin, the General Secretary position morphed from an administrative one into a de-facto power center. This struggle continued for a very long time, several Congresses battling between parliamentarian democratic forces that wanted to distribute state power between regional and local governments and centralizing pressures from Stalin and other bureaucratic factions. This battle was basically over after the 15th Congress, when Stalin's coalition captured all the important positions and swept the 'Left Opposition' out of government.

But even out of power, the respected voices of the anti-Stalin factions were able to organize and push for reforms. At the 17th Congress in 1934, many of the Delegates openly discussed reducing Stalin's powers, criticizing this new organization of government, and so on. Normal stuff for the Communist Party and a sign that there was a renewed push to return to decentralized power. Now, remember I said to put a pin in Stalin? Pull out the pin. Surprise, he was a grenade! Kaboom! Right after that Congress ends the Great Purge begins, wherein Stalin's NKVD basically wiped out the Central Committee, the Politburo, and any other rivals.

The result of this was an absolute consolidation of power under Stalin in the lead up to World War 2. Of the 139 Central Committee members elected at the 17th Congress, only 31 made it to the 18th Congress in 1939, and only 2 of those got re-elected. Not long after this, in 1941, when Stalin essentially took the office of Premier from the current office-holder as a way to unify the USSR under his leadership as war with Germany loomed. About a month later, Germany invaded. Stalin maintained his absolute grip on power until his death in 1953, after which an absolute fiasco of power grabbing caused the slow political implosion of the Soviet State that we actually continue to see today.

So how would you 'go about' getting into the Politburo after Stalin?

This is essentially no different from any other political career. In general terms, you need to be acceptable to your peers, useful to the leadership, and exercise the right balance of ruthlessness and restraint so as to advance the interests of yourself and your leaders without ending up on the losing side of a major power shift.

Politburo members worked part-time so they had other functions as well. Generally, they came from the state functions, but over time the number of central party leaders increased. The one distinction was the position of Defense Secretary, which saw a large number of Politburo memberships, and many leading Politburo members had served in the various wars.

The real problem with any discussion of the Soviet leadership system is the series of Purges that Stalin and others conducted to obliterate opponents. These purges would wipe out huge portions of the leadership in an area, sometimes replacing all but the entirety of the region's government and senior military leadership. Those who were swept up in a purge would be replaced, exiled, imprisoned, or executed. The number of vacant posts created by a purge allowed extremely unusual leaders to emerge, and many senior members of the Communist Party were born as destitute peasants, laborers, metalworkers, and so on.

This stands in contrast to the way party leadership behaved before the Purges, but you specified 'Post-Stalin' so you have to assume any leader in the Politburo was both ambitious enough to rise, while lucky enough to choose the winning side of all the purges and disasters. Stalin shuffled staff (not always violently) to find Leaders capable (or lucky enough) to succeed at the tasks he assigned them, so even being in his inner circle was not an assurance of continued approval.

So if you wanted the fastest possible track to Politburo membership you want to do one of two things:

First, the Patronage track: you want to develop a close relationship with a powerful leader in the Communist party so that, when they are choosing advisors, you are fast-tracked into a post or position on a committee. Family friends are excellent connections. If possible, head to a school for education in logistics or engineering, which are two fields that gave the Soviets a lot of problems, but do not require going abroad and potentially causing yourself problems. From here you should work to advance your patron while serving the power of whatever leader is currently running the show, especially by accomplishing whatever central planning goal the leadership has set. These successes elevate your status, probably get you onto the central committee, and from there you prove yourself so useful that they can push for you to join the Politburo. This is essentially the track that Kruschev took.

Second, the Military-Industrial track: you want to join the military at a young age and demonstrate bravery. Military success will allow you to move up into the Officer's branch of service, opening up secondary education into military matters. From here, again, you will want to look into logistical and engineering fields. Administrative capability is also favored. If you can accomplish something of merit and be picked out from a crowd then you may be fast-tracked for a position within the defense industry, or even as Defense Secretary, and this will assure you of a quick rise (as a vital component of the premier's inner circle) to the Central Committee and the Politburo. This is what you saw from people like Zhukov.

Many other Politburo members were just communists who served in good faith during the various administrations before Stalin, kept their heads down, and supported like-minded allies into positions of power. These folks don't make history books but they did exist.

In any case, the people most likely to end up in power were the ones who had supported Stalin from the beginning. Stalin was a leader of the most aggressive, most confrontational, and most authoritarian wing of the Communist Party, and there were a number of folks who had always been a fan of that even before Stalin was on anyone's radar. These leaders quite naturally rose up in the ranks as Stalin gained power, avoided the purges, and found themselves comfortably situated. One such example is Lazar Kaganovich, the guy who was Kruschev's patron. Lazar started off as an organizer before the revolution, got arrested, kept doing it, and then led a revolt during the revolution. He kept serving in leadership positions, dealing harshly with uprisings and dissent in the early days, which got Stalin's approval. Stalin ended up keeping the guy close his whole career and using him to rubberstamp some of his worst stuff, and Lazar stayed a devoted and loyal follower to Stalin. Nothing bad happened to him until Stalin died and Kruschev started de-Stalinizing the USSR. Guy lived until 97.