The most natural way to do this is after at least a couple of back/forth between you, wait for them to say something 'interesting' or 'funny' (you set the bar to your own ends but they must believe you genuinely find it interesting or funny), and at that point you have a legitimate reason to be more interested in them, they are interesting/funny... It's now not awkward at all, naturally they have given you a reason to want to know them better, so you then say something like "My how interesting/funny - what's your name?".
In a sentence, wait for them to give you a reason to want to know their name, then ask for it.
It must be "What's your name?" not "Who are you?", the latter sounds awful to be honest.