How to calclate Rb value for NPN transistor?
You generally should have a resistor on the collector (or from A to Vcc) to limit the current. Refer to your LCD datasheet for guidance on that matter, and assume transistor voltage drop is something like 100mV. Failure to include the resistor when one is required will likely lead to early failure of the backlight and/or transistor. Some LCDs have a suitable resistor for the expected supply voltage built-in, or have a place for a (fairly large physically) resistor on the PCB. There is a lot of variation between different products.
For the base resistor we typically try to drive it with about 1/20 of the collector current, so pick it so that base current ~= (5V-0.7)/Rb is about 1/20 of the backlight current. Say the backlight is 20mA you would want 1mA so Rb ~= 4.3K.