How can I play music while chatting in Google Hangouts?
What you need is an audio loopback.  Basically, what it does is it takes the output from one device and "loops it back" to the input of another device.  

You're in luck.  There's an App that does that and coincidentally, it's called "Loopback."

Macworld has a good article on it and apparently, it's rated quite high.