I have also looked into this. My sister and I use Hangouts exclusively for chat. She has no phone but a PC. She likes to work from her browser. I can send+receive messages on phone+pc. It works great for us to communicate instantly.
However since Google is taking away the chat part this month, we can no longer communicate how we do. I researched a little and someone suggested Telegram. It has a browser extension. So we can do the same thing as with Hangouts.
So I looked into Telegram to see if it works with Google Voice. I found some differing posts. You might try it yourself. It seems the majority of the posts that I read say it won't work with VoIP numbers but it doesn't hurt to try.
You can still use Google voice separately and use Telegram for just chatting via pc and phone. That is what we will do.