Watch videos of Waits playing piano. Listen to the style and if possible, watch his fingers play the keys. Take notes of things you see or notes that sound interesting. Also make notes about Waits' mannerisms and performance.
Get a chord chart of a Waits song you would like to learn. Learn how to play these chords with both your left and right hand. Keyboard style for modern rock or pop music typically follows the rule that you play a bass note or two in your left hand while you play the chord in your right hand. It is helpful to know some basic music theory, so you know what the notes of each chord are and how you can voice them -- for instance, which note is the top, middle or bottom.
Learn some jazz piano. Waits' song "The Piano Has Been Drinking" follows a jazzy pattern. Learn jazz chords and how they can be easily laid out on the piano. It might also be useful to learn some scales to know what notes you can use to improvise with while you play certain chords.
Listen to the Waits song you want to learn and play the chords along with it. Listen for times Waits improvises or plays a melody. Use your ear to figure out the notes and practice them.
Play the entire song along with a video or recording. If you want to write your own song in Waits' style, use similar chords or performance characteristics to make your presentation more like Tom Waits.