I think Stalinism doesn't necessarily follow from Leninism, but they are very much connected, especially in the context of the time.

To explain that, I'd like to start with somewhat arbitrary definitions of both, focused more on practical policy than ideology. I like to describe Leninism as "leap before you look" approach to socialist revolution. Sure, the final goal of communism is there, and socialism as a non-capitalist economic system of social ownership is a way to get there. It is just that for Lenin, or anyone in Russia at the time really, it was far from clear what socialism should be like. And it was very much flexible. Lenins' State and Revolution written immediately before October had a very nice, even utopian view of socialist society: democratic, worker managed, etc. However, it was very much modified, and likely had to be, to fit the conditions of relatively small working class, civil war, foreign intervention, initial industrialisation, power consolidation...

Stalin, in a practical sense, represents the acceptance of "socialism in one state" as a present condition likely to extend for some indefinite time in the future. It is also a response to what were largely considered as revolutionary retreats or even reactionary moves of the 1920s. Major practical aspects of Stalin could be summarised as: rapid, forced industrialisation and collectivisation, militarisation to preserve the "first state of socialism", and violent purges. All of those stem from 1920s, and in that case Leninism does very much set the stage for Stalinism.

Lenin once spoke of communism as being "soviet power + electrification", implying rapid and modern industrial development. However, during 1920s although industrial development did occur it was not considered rapid enough. Stalin, both for security, geopolitical concerns and to reverse the "backwardness" (in industrial terms) of Soviet Union, launched a program of rapid and forced industrialisation. Disastrous collectivisation of agriculture was to provide both resources, export goods and work force to make that possible.

The idea of Soviet Union as a besieged bastion of socialism, permanently in danger from intervention was not as unreasonable as it might appear in the West. After all, western powers did intervene once, anti-communist rhetoric was very strong, and the idea tied naturally into ideological bases of socialism. Bourgeoisie would be ready for anything to preserve its class power. Marx himself, writing for example about Paris commune (Civil war in France being the best example) wrote very emotionally and persuasively about the counter-revolutionary violence.

As a side not, Trotsky is usually considered as a great "road not taken" alternative to Stalinism, one that would greatly reduce violence and repression in USSR. I can't go through my copy of Revolution betrayed now, but I'd disagree. Major difference was the idea of imperative to spread the revolution. On the rapid industrialisation and collectivisation, Trotsky criticised Stalin for not being rapid and violent enough, if anything.

Purges were a normal way for a Party to function. It would expand, and than periodically those deemed not revolutionary enough, or not worthy of membership were purged. It was a way to have a massive Party but also to keep it reasonably homogeneous and revolutionary. Purges were generally not violent, Great purge was an obvious exception. But stage for it was also set in 1920s, especially with NEP (New Economic Policy). It was considered by many, some old revolutionarys and by many idealistic young people, as a retreat or even reaction. It preserved class distinctions, inequalities, "exploitation of man by man". At the same time, much of the middle and high management possitions were permanently filled by caders from Civil war. Those were often poorly educated, and there was little prospect for newly educated young coming into workforce to replace them. Until they were purged, that is. Therefore, I consider the Great purge in large part a popular movement. I've wrote on that before and while I don't have a single author or a source to quote on that it does sort of come through from some readings, but more on that in a linked post. Some of most influential sources for what I'm writing here are listed in that comment as well, so I won't repeat them.

So to try and summarise and conclude: Leninism does not necessarily lead to Stalinism, but in the context of the Soviet Union in 1920s it does strongly suggest it. In the 1930s it was, if anything, a Great return to revolution, not a retreat. That view developed among those ousted from power (Trotsky again as most well known), and became more prominent later, after the Second world war and more so after Stalin died. Many features of 1920s and especially NEP were seen as a retreat however, and Stalinism was in significant part a response to that.