How do you write a lyric poem?

A lyric poem is a personal poem. Just write whatever is on your mind. Tell your story.

Express your concern. If you want to imitate a writer, just create a poem like the one you are reading. For example, look up "Sea Fever". Think of something that you feel compelled to do.

Then write a poem like "Sea Fever". If you want to write a Sonnet, read a Shakespeare sonnet, then write one like that.
Lyric poetry is what you probably think of when you think of a poem. Songs are lyric poetry, also. Click on the Related Questions to learn what sorts of lyric poems you might write.

To write good lyric poetry, you must have a good sense of rhythm and rhyme, and be able to fit your Words into the proper format without resorting to stilted phrases or forcing the emphasis onto odd syllables.

Pick a topic and brainstorm: just write down any images you think of, any rhymes that come to your mind, and any Words that are associated with that topic for you. Once you feel that you have a big enough "pile" of images and Words, start looking for connections. Join the bits together in different ways - try one rhyme, then another until you find a pattern that works for you.

Then, start editing. You will need to condense your poetry down to it's most concise form. Dump anything that takes away, even slightly, from the images you want to convey. Make certain that each Word is precisely the one you need to invoke whatever emotion you are aiming for in your reader (or listener). This step will take longer than writing the images down, because a poem must be distilled down to bare image and emotion.