The good news is that you don't need to kill your existing grass or use any chemicals.
If you hire someone to install the plugs, it can help for them to remove some of the grass around the plug. However, it's not necessary.
I've installed Zoysia two ways.
1. I killed existing grass with plastic tarps and then installed the Zoysia. It didn't matter if every last thing was killed or not. The Zoysia will take over.
2. I installed plugs into existing grass.
Both worked. You have to be patient, though, as it can take a few months before you see the difference.