Good luck. I, at least, have always been able to sing while playing -- one of the things a choral conductor simply has to be able to do -- but I have never been able to give full justice to the interpretation of either the vocal line or the piano (or organ) accompaniment. Indeed, I honestly can't say that I've ever known anyone who could.Particularly for something like lieder, it is important to realise that one is not really dealing with a vocal line with an accompaniment -- such as might be found in a folk song (something which I have been able to do successfully!) -- but two instruments which are genuinely equal, and must be equally virtuosic. Not that that is always clear on recordings or even recitals (one tends to hear about the singer, not the accompanist), but it is clear in the music, and if it is not true the result is either a catastrophe or simply horrible.
I've never dared to perform lieder like this, though, even though technically I could. Somehow, I'd feel something is missing: the intimate communication through music between singer and pianist. Maybe I'm wrong.