Select a simple accompaniment -- one that is not too rhythmically complex. Block chords are best to start. Later, you can elaborate with arpeggiated chords and a simple bass line. Good instruments for accompaniment include piano and guitar.
Become familiar with the musical structure of the song you wish to play and sing. Many songs consist of a refrain and one or more verses or strophes. Or, the song may have several strophes, but no refrain. For the latter type of song, the melody and accompaniment tend to be more varied and complex.
Learn the accompaniment for the refrain first, practicing repeatedly until you can play it almost from memory. Having mastered the accompaniment, your mind is free to focus on adding the lyrics and singing expressively.
Try humming the tune along with your accompaniment. Doing so allows for greater concentration on getting melody and accompaniment to synchronize well.
Practice singing the lyrics of the refrain only, without accompaniment. This kind of singing is known as "a cappella" and is a good way to secure the melody in your mind. If you're just getting started, you might wish to listen to a recording of the song or have someone play the melody on a piano.
Add the accompaniment as you practice singing the simple refrain. Remember that singing expressively is important, so the better you have mastered the accompaniment, the more artistic freedom you have for interpreting your song.
Practice the verses of your song in the same manner -- humming first, then singing the lyrics along with the accompaniment.
Play the entire song, both accompaniment and vocals, slowly at first. Although you repeatedly sing the song at a slow tempo, be sure to keep the tempo and rhythm steady.
Repeat the steps in previous sections as needed. There may be certain passages that are more difficult than others, so be sure to give them adequate attention and extra practice.
Play and sing your song at a progressively faster tempo, until you reach the natural speed of the song as you wish to interpret it.