I am looking at this(instructable with circuit picture) tutorial for making a self-balancing robot.
However, I am confused about how the battery connects to the arduino.
Well, it seems he has attached a 9V battery through a voltage regulator, to the Arduino.
Which is bullshit.
You can put the 9V directly into the barrel-jack of the Arduino. You actually should. Putting 5V on the barrel jack is not within the specs. (https://www.arduino.cc/en/main/arduinoBoardUno)
I know the 12 V must go to the motors to power them yet the diagram shows wires connecting to the Arduino through some kind of green device. I assume this connects to the power supply . However I do not have this green device and I don't know how to connect to the power supply.
Well, the motors only get 9V in this diagram (which might work neverthless). But bear in mind that 9V batteries won't last long.
I'm not sure what you mean by "Green device", the HC-05 or the L293D?
Anyway, the L293D are quadruple high-current half-H drivers, which basically is a way to switch 12V on/off/invert based on the 5V inputs (from Arduino).
You could do this with some circuitry, but I suggest picking up a "motor driver (arduino)"
How should I get a charger cable and solder on the battery wires to
You can't charge regular 9V batteries.
Is the connection to the Aruino necessary?
The specific connection he shows? No.
Power to the Arduino board? Yes.