The best starting point when writing a critical appreciation of a poem is to determine what the author's message or theme is. Once you determine this key point, the rest of the paper falls into place much easier.

I tell my students to next look at the author's tone. What is the author's attitude toward the subject of the poem? How do you know? What words do they use to convey that tone? You can then discuss how the tone is important to the theme of the poem.

It's important to then look for literary elements you're familiar with. Do you see a simile? What about great imagery? Is there an example of personification? Once you find these, the key point to remember is that the author made these poetic choices for a reason. How does that simile contribute to the tone or theme? How does the imagery utilized bring deeper significance to the tone or theme?

For example, in the poem "Because I could not stop for Death –," Emily Dickinson chooses to personify Death. Why does she make that choice? The image provides a way for her to discuss the transition into the afterlife as though she is a passenger in a carriage with Death. By doing so, she is able to reflect upon the life she's lived in light of the knowledge of death that all people face.

In the poem "Stopping by Woods on a Snowy Evening," Robert Frost uses imagery to convey the peaceful scene before him as he pauses in snow-filled woods on a dark evening. Why does he use imagery there? The peaceful imagery he utilizes contrasts with the "But" at the end of the poem; he has "promises to keep" and "miles to go" before he sleeps. He is in a hurry and doesn't have time to enjoy this peace often.

Each choice the poet makes is for a specific purpose. It's important, therefore, to not simply identify those literary elements but also to analyze how each one brings deeper significance to the author's work.

The above answers give good insights, but as you can imagine, there is no one way to appreciate or analyze a poem. In other words, there are many ways in which to approach a poem. In light of this, let me give you a few different ways. 

First, if you focus on the mechanics of a poem by examining poetic devices, you should also ask yourself what these devices do for the meaning of the poem. For example, the meter of The Tyger by William Blake reminds the reader of the incessant pounding of a hammer on an anvil. In this way, the poetic elements add to the meaning of the work.

Second, you should also focus on the historical context. For example, the Roman poet Horace writes ode 1.2 in the context of the flooding of the Tiber River. This act can be seen as a bad omen of what is to follow, or he can interpret it as the signaling of a new age with the rise of Augustus. In this way, he is saying the poet has a lot of power to interpret events in poetry for the general populous. 

In conclusion, to analyze poetry, use as many tools and ways of reading as you can. That is critical.  

A critical appreciation of a poem requires of one to analyse the poem as a whole and critically provide insight into the elements which make up the poem, such as diction, imagery, structure, rhyme, rhythm, the overall message or theme of the poem or the purpose of the poet. One should also be able to determine the context and setting of the poem and its relevance to the period in which it was composed and how it relates to the current context. Furthermore, one has to mention how and why the above-mentioned elements are either effective or not - this is the 'critical' aspect of one's discussion.

Obviously one has to apply the structural requirements for an essay - i.e. an introduction, body and conclusion. Furthermore, the requirements for proper punctuation, grammar and language, should be followed.

One needs to mention in the introduction the purpose of the essay through a thesis statement,discuss the various elements of the poem in separate paragraphs and conclude by either restating the thesis and the purpose of the essay or in some way requiring of the reader to make an assessment of your attempt or both. However the conclusion is worded, there should be a definite indication that the essay has reached its end.  

In this context, “critical” means paying attention to the elements of construction – rhyme scheme, meter, stanza arrangement, imagery, etc. – that give the poem its balance, beauty, and effectiveness.  Writing an “appreciation” requires a dissection of the way the poet has achieved his/her effects, and should be constructed like any essay – introduction, body, conclusion, paying particular attention to those elements that give poetry its signature – succinctness, “concentrated word magic.”  If other poems by the same poet are known, you may discuss how this poem differs from or emulates the poet’s  “normal” style; in a longer appreciation, you may also discuss the “age” or “style” of the poem – Romantic, Victorian, etc. -- and you might discuss the generic style – sonnet, ode, etc.

How do I write a critical appreciation of a poem? What are the components?

It depends on what your professor stipulates as the length for the final draft - but if this is a 2-5 page typed paper, I would say focus on three categories of criticism.  Think about this as your typical 5-paragraph essay - most good analysis is organized into three chunks - but instead of only devoting one paragraph to each category - feel free to write more.

As far as the "parts" of critical appreciation go - you have almost endless options with this.  You can approach it from the "What are other people saying?" route - where this paper is actually more of a research paper than a personal analysis - but your own assessment is intertwined in how you present your information.

If outside sources are not required than likely your professor wants only personal analysis.  Here, you rely on what you learned in high school English: literary elements and rhetorical techniques.  What is this poem trying to say and how effectively does it say it?

I'd say good critical analysis combines any two of the following: tone/attitude, theme, and literary techniques (things like figurative language, point-of-view, irony, symbolism, imagery, etc.)  You can look at one main theme of the poem and analyze 3 different techniques used to portray this theme.  You can analyze the overall tone of the poem and look at the different themes expolored through one tone.  Do you see how the combinations can be played in different ways?

So again, while the exact structure of the 5 paragraph essay may not be quite enough, use it as the framework to explore the poem in 3 categories - just make sure the 3 categories all revolve around a common focus (thesis).  The order of presentation (no matter how many paragraphs are devoted to each category) is best as follows: 2nd strongest point, weakest point, strongest point.

Hope that helps.