It's hard to see for sure, but that looks a lot like standard NM wire. I've seen folks run the neutral to a screw in the box to make it appear grounded. While it tricks the tester, it is not a safe configuration. I would check for that. I've also seen folks splice NM behind the box to cover up that it is really being fed by old knob and tube or other cloth covered wire. If you're not sure, I would take the extra time to check. I found charred/burnt wire in this configuration in my house.
You'll want to put the GFCI outlet in the first outlet in the series. All outlets after will then have GFCI protection and can have three prong outlets installed, with label. Please check your local codes though!
You'll want to put the GFCI outlet in the first outlet in the series. All outlets after will then have GFCI protection and can have three prong outlets installed, with label. Please check your local codes though!