"Do you know this boy?"
Translation:Znacie tego chłopca?
Yes. Will be a lot better for a situation when the boy is older, like 13+ (hard to set the border).
you need accusative after "znać". Accusative form of chłopiec is "chłopca".
chłopiec is masculine animated noun, so it has accusative=genitive. Only neuter and masculine not animated nouns have singular accusative=nominative.
"tego" is a form of "this". "tamto" is a form of "that".
"tamto" is neuter Nominative or Accusative. "tego" is either masculine animate Accusative, masculine Genitive or neuter Genitive. Here it is masculine Accusative, "znać" takes Accusative and "chłopiec" is masculine.
I advise you to save the declension table somewhere: https://en.wiktionary.org/wiki/ten#Polish
It is for forms of "ten", but forms of "tamten" are almost identical: you just put tam- at the beginning. The only difference is feminine Accusative, which is "tę" for 'this' but "tamtą" for 'that'.