All you need to know is described on this page: Remote Debugging on Android with Chrome.
I tried to set up remote debugging on my Samsung Galaxy SIII. My operating system is Windows 7. My desktop Chrome browser didn’t discover my device.
I encountered one big problem in the step: “Discovering devices in Chrome”. In “Troubleshooting” part they suggest to install OEM USB Drivers on the Android Developers’ site. And I did everything what they said…. But my desktop Chrome browser didn’t discover my poor Android.
So, in conclusion, make sure to install Samsung Android USB Driver for Windows.
That’s all! My despair was gone. 😀