I have a lot of experience accompanimenting to singing. And I can say that two things are the basic of successful accompaniment - your theoretical preparation and your experience. By the main method of play for accompaniment to yourself are long-holded chords,smoothly transferring one in other and top part of which can make some melodious line. Anotherwell known method of accompaniment - is to play with right hand metrical harmonic chords, which repeat the rhythm of the chords in the left hand, or ( in more slow tempos) are more various rhythmically.

It's better to start with simple long chords, gradually making harmony and rhythm more improved. Then it is possible to add the small melodious insertions while there is pause in the melody of song or long note.

I should also mention the importance of function of bass in accompaniment - it must always be heard and sound deep, enabling to the performer to lean against its strong beat.

Glad if I able to help you.