"Do you like my clothes?"
Translation:Liker du klærne mine?
It just doesn't make sense. "Liker du klærne meg?" sounds as unnatural as "Do you like the clothes my?"
The proper way of expressing this is "Liker du klærne mine?" Klærne mine is the most natural way of saying "my clothes," but mine klær is also fine to express more emphasis on my clothes and not someone else's clothes.
"mine" is the possessive pronoun, and equals "my" or "mine" in English, whereas "meg" is an object pronoun and equals "me". The only difference is that the order is a bit different in English.
Klærne meg = Clothes me
Klærne mine = My clothes / clothes of mine
And as Like_5.1991 says, you could even say "mine klær", which gives a slightly different emphasis.