I'm trying to remember how I learned to do what I do in this instance.
I know I always have my right hand "over" the same chords that my left hand is playing.
Generally I sing a line and then play some sort of "answer" fill. Only in the past few years have a learned to actually play more than a chord while I'm actually singing.
Start by just playing the notes in the chord one at a time, in time of course. Experiment to see what other notes out of the chord sound good when you play them. You will often find that the flatted third of the chord sounds good, perhaps sliding from it to the third. Then experiment with the sixth note of the chord. It gets easier the more you do it.
Also, listen to a recording of the song you are playing, hear what the "signature" riffs are and learn to play them. The intros are especially important. I almost never use a factory intro; instead I play a line from the melody of the song, or actually try to play something close to the original intro.
Now if I had to actually play much with the left hand, I would turn into a blubbering idiot. Hope this helps a little.
P.S. I just saw Chas post and he is exactly right, although as I mentioned, I'm working on actual comping more. I don't have a problem coming behind someone else, but it's a lot harder to do it while you are singing.