One way to get some rhythm into your right hand
Here is how you can get some busy-ness and rhythmic sounds into your right hand while your left hand is doing it’s thing: Break up the chord notes UNDER the melody – sometimes from the top down and sometimes from the top up, and sometimes syncopated and sometimes steady. And sometimes just play the chord — in other words, use lots of variation so it was stay interesting to your listener.
Watch this short video to get the idea: