That would be मुझे आपके परिवार से मिलनता पसंद है. You could also use तुम्हारे परिवार or तेरे परिवार depending on context/level of familiarity with who you're speaking to.
I think part of the confusion here is just that आपके and अपने sound kind of similar. The difference is very important though!
Is the "aapne" correct here? In the Hindi sentence, the subject is NOT "I", so would it not be "mere"? The sentence is more literally translated as "Meeting my family is pleasing to me" (like in Spanish, for example). So the subject is the 3rd person singular "meeting/to meet".
That's a convenient way to translate it for learning purposes in some sense, but for most practical senses, मुझे is the subject-- What is called, in languages ranging from Hindi to German to Georgian to etc. etc. etc., a "quirky" or "non-nominative" subject. Part of the evidence for the idea comes specifically from the fact that you get अपना in sentences like this, though there's a number of other sorts of evidence for it. In any case, regardless of the analysis in linguistics, for this sentence, अपने is what should be used.
अपना is sort of a funny thing, at least from an English perspective -- it's the possessive form specifically for when the possessor is the same person as the subject. This is clearest with examples like this:
आमिर उसकी किताब पढ़ता है -- Amir reads his/her book; the book belongs to someone besides Amir
आमिर अपनी किताब पढ़ता है -- Amir reads his (own) book; the book belongs to Amir.
A loose way to translate अपना would be "self's". so in my example, "Amir reads (him)self's book", or in the original example above, "I like to meet (my)self's family". (Obviously "myself's family" is not natural in English, but it's a way to think of the way अपना works)