The word “Philosophy” literally means love of wisdom, so one answer might be questions asked strictly out of the love of wisdom or knowledge –questions whose primary purpose is not one of practical utility. I don’t favor this definition, however, because I believe philosophical questions have a lot of utility, although not always in an obvious manner.
Another definition is hinted at by Socrates, in Plato’s Phaedo, when he says that he studied science in his youth but found it inadequate because it only answered “how” questions and not “why” questions. Taking this as our guide, we might say that philosophical questions are those more concerned with “why” than with “how”.