Not too wrong. I'll re-write it and then explain:
"On my way to the underground today, I heard a father talking to his kid whilst they/I were/was passing a yard. He said "look at your face, it's just like a dirty as.. What the hell do you want to do, go begging?" My first thought is 'is that his kid'?
Your meaning is pretty much clear, just be careful of swapping tenses. What you mean to say, I think, is that you were heading to the underground right? In the story you are describing, even though it is in the past, you are describing a continuing action. So, because you were walkING, the father was also talkING to his child, right? Just be clear, too, on who was passing the yard: was it you or them?
Let me know if you need further explanation!