How does Milan Kundera portray communism through characters and events in The Joke?

Kundera makes an argument in The Joke that humor is complex and intricate.  It is complex and intricate because it can help to provide a frame of reference for our world and our place in it.  Such complexity is repudiated through Communist ideologies. In this regard, Kundera is making a statement that individuals who seek a world of simplicity are threatened by the multi-faceted understanding of humor.

In Ludvik's initial attempt at humor, there is a depiction of Communism as this singular reality that fails to understand the intricacies of humor.  When Ludvik writes, “Optimism is the opium of the people! A healthy atmosphere stinks of stupidity! Long live Trotsky! Ludvik," it is a statement of how complex humor and satire can be.  "The joke" becomes a statement that compels individuals to think.  It lies beyond the neat and simplistic constructions of Communism.  The fact that Ludvik is rejected as a result of his joke, a consequence of complex thought, is how Kundera portrays Communism as a reductive force.  It seeks to remove the intricacies and complexity of consciousness in order to substantiate its own power and control.  Ludvik is a victim of the historical reality in which people seek power at the cost of intricacy of thought.  He himself becomes the victim of his own joke as he fails to see the multiplicity of outcomes of his actions.  Communism and human beings who support it are seen as elements that seek to affirm simplicity in order to substantiate their own power.   

What is Milan Kundera's criticism of Czech society in The Joke?

If you want to put a simple political take on it, The Joke is a reaction to the eventual takeover of Czechoslovakia by Communism.  Kundera continually combines the personal with the political during the entire book.  All of the characters in the story are victims.  What are they victims of?  "The joke history has played on them."

For example, Ludvik always seems to be having some kind of tragic experience, a typical result of Communist overreaction.  Ludvik's and other characters' devastation is certainly connected to the Communist takeover of Czechoslovakia. 

The political bleeds into the cultural in regard to devastation as well (again blamed on Communism).  There is an attack on Christianity and even an attack on folk music!  It is the Communist way to squash the culture of a country in a form of fake unity.  Kundera isn't afraid to show this with his dark humor.  Kundera shows through Ludvik and others that there is now a youth culture completely devoid of Czechoslovakian ways.

What is the personal "joke" of the book (and how does it connect to politics)?  Well, Ludvik is obviously trying to twist the knife with his ex-lover by sending this note:

Optimism is the opium of the people! A healthy atmosphere stinks of stupidity! Long live Trotsky!

The irony is Ludvik's "joke" is on him because the Communist authorities intercept it and don't appreciate it, forcing him to work in a mine.  Such is the reward for Ludvik's devotion to the party.  So, therefore, it's not a surprise that this book was banned after the Communist takeover of Czechoslovakia.  It has a dark, even nihilistic, tone as can be noted from this quotation by Ludvik:

Everything will be forgotten and nothing will be rectified. All rectification (both vengeance and forgiveness) will be taken over by oblivion.

In light of all the previous evidence and ideas, it is important to recognize Kundera's own remarks on the subject.  In 1980, someone on TV remarked that his novel was “a major indictment of Stalinism.”  Kundera was really quick with a retort: “Spare me your Stalinism, please.  The Joke is a love story!”  This shows that Kundera meant his novel to be not just a mockery of the Cold War Era, but also a story in itself.