Here's a couple of me playing and singing:
I don't have any brilliant tips for it. I just learn the piano part until I can play it cold, then start adding the singing. It's just like putting two hands together: learn each separately, put them together, start slow, work your way up to tempo, etc. The hardest part for me is remembering lyrics (something I'm bad at, piano or not), so I work on that away from the piano a lot too.