The common expression is.. I want to get to know you more. This is assumes you know that person already (but only a little bit).

When you first meet someone you should say, "I'd like to get to know you better". <<< This is correct.

당신과 친하게 지내고 싶어요. The direct translation is : I want to stay friends with you. (not become friends and not to get to know someone better = get better acquainted)

In that case.. you say.. I want to get to know you. "I want to get familiar with you" is not natural. And because it is the first meeting.. a more natural expression is.. I'd like to get to know you. This is less confronting than "want". Sometimes, exact word translation doesn't really match - as you probably know. Hope that helps ^^

I'm not sure if that is a direct translation, but that's not an English expression... Maybe you are trying to say, "I want to get familiar with you."??