I've always wanted to do this. I made a screen recording of me writing a poem from beginning to end. It took about ten minutes.
What I did was cheat. Well, I began with a poem by the great Swedish poet Aase Berg. I copied it into Google Translate and then I translated back and forth through several languages. I tried to choose unrelated languages in order to get a "translation" that was very different from the original.
Then I copied the raw translated poemstuff into Word and began editing, cutting out what didn't seem interesting and sculpting a poem out of the material that I had to work with.
At one point, I didn't think I had enough poemstuff so I tried doing some of the process again, except with the new edited poem as far as I'd got with it. I put it into the Spoonbill.org N+7 generator (which finds nouns in the dictionary which come from 1 to 15 entries later than the nouns in the seed text), then I stuck that lot into Google Translate again and did some "translation broken telephone."
Then I again pasted the result into Word and considered how to add to my emerging poem. I ended up not using any of it except one line.
A bit of fiddling—reordering words and moving lines around—and then the poem was done. Except for the title. I stuck the poem into the N+7 generator to find some new words. I found "Bramble" which I liked as a title and so the title it became.
I'd love to narrate my thought process—the decisions I made as I went—but perhaps just watching the editing in action is enough by itself. By the way, I said this was in real time. I actually sped it up just a smidgen so that it moved just a bit quicker in order to keep it interesting.
A note on the music: I made the music by a related process. I took a bit of Vivaldi—a recording of a vocal solo accompanied by strings and translated it into MIDI notes using a sequencing software. Like translating languages, the process is inexact, especially if you deliberately mess with it which I did. I made separate "melody," "harmony" and "drum" translations and then played around with the result to make the final music. (I chose the instruments, added in a bit of the original recording much processed and so on.) So it is a perfect analogue to the process I used with the poem.
Oh and why did I use this process on the poem? It's an interest way to generate raw material to work with. Of course, depending on what you start with, you get different results. Some very abstracted remnant of the form and tone is maintained. If I had chosen another poem, or another poem, I would have got very different material to work with. Also, something of the original poet's style was vaguely in my mind, though my result isn't at all like hers. I'm not saying this is the best example of poeming known to lifeforms around the universe, but I think the result is interesting and the process even more so.