Your main aim in writing a critical essay about a literary work is to address some issue that would not be immediately obvious to someone who has read the book. Thus, a plot summary is not a critical essay, because it covers ground that would be familiar to any reader. What would count as a critical essay would be analysis of such things as what works influenced the play, the differences between the French and English versions, the relationship of the play to its historical context, or the use of literary devices in the play.
Your essay should start out with some sort of an analytical thesis. So, for example, you might talk about how Lucky's "thinking" is might have been influenced as much by Flann O'Brien's short essays as by earlier philosophers. Once you have introduced your main claim in your first paragraph, you should write a transitional paragraph pointing forward to how you intend to support your claim. For example, if talking about O'Brien's influence on Beckett, you would state that you would first provide evidence for Beckett being acquainted with O'Brien's work, then do a close reading of the relevant works by O'Brien, and finally present those elements in Lucky's speech that show this influence. Your conclusion would clarify just how the body of your work proved your initial thesis.
A critical essay analyzes a piece of literature, such as the play Waiting for Godot. Rather than summarizing the work, a critical essay provides an analysis of what the work means and how the author conveys its meaning. In the case of Waiting for Godot, the writer of a critical essay would consider the meaning of the play and the meaning of the characters' actions (or inaction).
The essay begins with a strong, argumentative thesis statement that includes what the author is trying to prove. It usually relates to the meaning of the literary work. Each body paragraph, which begins with a topic sentence, includes sub-points about the theme of the work and includes relevant quotes from the work, including a discussion of the quotes. For each quote, the writer of the essay should consider the literary devices the author uses, such as figurative language or imagery, and should relate the quote to the themes of the work. The writer should also stay in the present tense when writing about a literary work and cite all sources of the quotes he or she uses with the proper format (generally MLA format). The writer should conclude the essay with a broad statement of how the work relates to life and what questions the literary work poses about life.