Joy stick Implementation on PID controlled balancing robot
I'm not 100% sure what your expect problem is … I assume you can read left and right on the joystick (very similar to up and down).

To make a left movement, you have to either stop the left wheel and rotate the right wheel forward (so you make a left turn), or if you want an even faster movement, move the left wheel backwards and the right wheel forward. 

For a right movement, do the opposite.

(actually having a different speed for the left and right wheel already causes a turning movement).