How do I write a thesis statement for "Dog's Death" by John Updike? Do you have suggestions for writing a literary analysis of the piece, too?

Updike's 1958 poem on the subject of a puppy's death is filled with pathos without lapsing into obvious sentimentality or a maudlin tone.
A good way to approach an analysis of this poem is to consider Updike's techniques. Look at how he arranges the stanzas and count the lines in each. Analyze the rhyme scheme; is it regular or irregular? Once you've determined that, think about why that might be appropriate to the subject.
What words are repeated? Why would they be?
What is the speaker's tone, and what words contribute to creating the tone?
How does imagery function in the poem? Are any of the images symbolic?
Remember that modern poems are often meant to observe; what might Updike be observing about the role that pets play in our lives?  What attributes are ascribed particularly to dogs, and how is that at work in the poem? How does the puppy's young age (i.e. not yet housebroken) add to the pathos?
Here are some literary terms that you should consider in writing an analysis of this particular poem: theme, consonance, end rhyme, slant rhyme, symbolism, imagery, quatrain, repetition, and tone.
When writing a thesis statement for a literary analysis, it is a good idea to write a closed thesis to limit the scope of what you will analyze. Acknowledging the theme in the thesis is also recommended. Here is an example:
Updike utilizes symbolism, imagery, and varying rhyme schemes to examine the relationship between people and pets and the emotions evoked by the unexpected death of a family dog.
Note that the thesis statement identifies a theme and three particular techniques that could be the focus of the essay's development.
Good luck with your analysis!