Can I get iPhone messages through my Mac? I am wondering if there is any way I can get my iPhone messages to come in through my messages on my Mac computer?

My iPhone is not working at the moment and has to get the battery replaced so it won't turn on.
Messages.app on OS X can sync messages from iPhone your to the app. This only works when your phone is on the same Wi-Fi network. 

So if your phone is getting repaired, you will only receive iMessages (messages sent font an iPhone/iPad/Mac to another iPhone/iPad/Mac), and not SMS texts.