The questions are an internal dialogue, so you may punctuate like dialogue:
I keep asking myself, "Who am I? Where am I from? Where am I going?"
Alternatively, you can make the conscious choice to break the colon/capitalization rule:
I keep asking myself: Who am I? Where am I from? Where am I going?
Or, you can revise your first clause so that it can stand on its own:
Unanswered questions keep swimming through my mind. Who am I? Where am I from? Where am I going?