Arduino LCD Brightness
The back light of the LCD is just an LED connected through a resistors on the A (15) and K (16) pins on the display. The LED doesn't require an external resistor since according to the datasheet there is a build in resistor on the board itself. If you aren't using any external resistors and applying a constant voltage of 5V and the screen is still dim than this means that the led is most likely dead and you would need to replace the back light panel.

If you are talking about the contrasts of the display. Than I would recommend you to hook up the contrast pin to a PWM pin on the Arduino and by using analogWite to tune the display in order to get the desired value of contrast.