Transistor: Mix both GND
The circuit you show in your scheme will not work. Chances are that the transistor is damaged. For a transistor connected as a switch, it is normal to place the load in the collector circuit, as I show below







simulate this circuit &ndash; Schematic created using CircuitLab

This is the basic way to connect a NPN transistor in the output of a microcontroller to activate a load.

When the output pin is in high, the transistor will conduct, feeding the load, in this case an LED. The value of the base resistance depends on the supply voltage of the microcontroller and the maximum current that support. The value of the collector resistor depends on the source Vcc, and the load that would like to activate. 

This way, you can share the GND connection between the microcontroller and the power supply to the load.