Before you start setting your poetry to music, analyze its structure. For songwriting purposes, you should pick a piece of poetry that has a consistent structure. It should have verses that are easy to tell apart, and the number or syllables in a line should be similar in each verse. If you write blank verse, go through and divide the poem into smaller sections that could be used as verses.
Sometimes it's easier to start writing music with harmony, not melody. Go to the piano or get out a guitar and pick a sequence of chords that you think expresses the mood of the poem. You might use minor chords, diminished chords or major seventh chords for a sad song, or you could use major chords and dominant seventh chords for a more upbeat song.
When thinking of chords doesn't give you inspiration, you might want to start with the melody. Sit down with an instrument and look at the first few lines of your poem. Try humming them instead of speaking them. Think about which syllables are strong and which are weak. In general, it works better to put strong syllables on strong beats (beats 1 and 3, not 2 and 4).
Whether you start out with a melody or with a chord progression, you'll need to combine the two. If you started with a melody, write it down on staff paper if you know how. If you don't, you can use your instrument to experiment with chords that sound good with the melody.
If you started out with a chord progression, analyze the chords. Find out what notes are in them, then use these notes to construct a melody. Try to give your melody an arc or contour, rather than hovering around the same note.
Once you have a musical skeleton for your song, expand it by writing other sections. You may or may not have a chorus, but if you do, you should make the chorus sound different from the verse. You may also want to write an intro, an outro or a bridge (a section of music that ties the chorus into a new verse).