It largely depends on what year are you living in and how old you are at the point when you start having the dream. There was just a single leader of the USSR born in the USSR, and that was Mikhail Gorbachev, born 1933 (strictly speaking there was also acting General Secretary Vladimir Ivashko... for 5 days).

Stalin, for example, was a revolutionary, then a commander of the Soviet army during the war against Poland, and eventually rose to power by making allies (Kamenev and Zinoviev), eliminating his enemies (Trotsky), and then eliminating his allies (Kamenev and Zinoviev). You could follow the same route, but you'd have to start before the USSR was even a thing. Nikita Khrushchev started his career as a political commissar in the Red Army. Post World War 2 he sort of made a name for himself in rebuilding the Ukrainian economy. When Stalin died, a political vacuum came about which Beria intended to fill. He was stopped dead in his tracks, arrested and executed. Khrushchev became General Secretary. Therefore, you'd need to climb the ladder at the right pace, be in the right place at the right time and have the right allies.

The previous paragraph is a very brief and perfunctory overview of the careers of these two men, but it already shows two very different approaches: rising through fire and blood, and chaos (chaos is a laddah, to quote a certain person) and rising through a more or less regular political career.

If you want to climb to the top, you would probably need to follow in the footsteps of Gorbachev. He was born to a peasant family, in no way related to the political elite. You would need the study at a good university (such as the Moscow State University that Gorbachev graduated from) and join the Communist Party as soon as possible. Then, you need to start climbing within the structures of your Regional Committee with your short term goal--the Communist Party Central Committee-- in mind. Whatever you do in your Regional Committee should be aimed at getting you into the Central Committe. Gorbachev got there at age 40.

Now you can start working on your long-term goal, i.e. becoming a member of the Politburo, which was the actual supreme authority of the country. You should probably be very active there and most definitely you need to make sure that the GenSec appreciates your talents and hard work (as Gorbachev managed to do with Yuri Andropov). After that it's a waiting game for the GenSec to die off. Once this happens, there will be a power struggle, generally a bloodless one, where again you'll need to choose your allies wisely, so that they don't backstab you. Gorbachev lost in his first bid for power after Andropov died, and only succeeded after the death of Chernienko.

Therefore, if you're a young Soviet citizen born after the USSR was established, your best (and only) plan is a quick climb of a long political ladder backed by proper education and a certain political cunning.

If you're interested in reading more about Gorbachev, you might want to read 'The Dead Hand: The Untold Story of the Cold War Arms Race and its Dangerous Legacy,' by David E. Hoffman. The book is not about the life of Gorbachev, but about what his (and Reagan's) coming to power meant for the world. For Stalin, I reccommend 'Stalin: The Court of the Red Tsar,' by Simon Sebag Montefiore