Electronic – LCD : what is the material inside

Line width is the number of characters that can be displayed on one line. In the description section, the datasheet says:

The maximum display size can be either 80 characters in 1-line display or 40 characters in 2-line display.

As Olin says, the datasheet is pretty terrible. The standard of display datasheets seems to be quite low in general IME.

It states Vss = GND, and Vdd - Vss max is 5.5V.

The LCD driving voltage looks to be internal so don't worry about that.

The Backlight doesn't need a negative supply, the A and K are for anode and cathode of the LED. It states either 40mA or 120mA for the BL current consumption, depending on the module type - in the mechanical specs table it specifies either no backlight, an EL backlight or LED backlight. I'd assume the edge and array in the electrical specs refer to the EL and LED respectively.

So if we assume your model is the EL, you need 40mA to drive it. You will need to confirm this and adjust the below workings as necessary if it's the 120mA version - if you don't have the info on the package it came in there is probably a code printed on the PCB somewhere. Either way it's safer to assume it's the lower current version to start with if not entirely sure.

You need to know the Vf of the LED, which is not clearly stated, but VB/L = 4.2V is given which is possible for a white LED.
You could measure the drop if you have a multimeter with diode setting available.
So, assuming the LED Vf is 4.2V, you are running the Vdd from 5V and need 40mA for your edge LED, you get (5V - 4.2V) / 0.04A = 20 ohm resistor needed between Vdd and A (K to ground)

For Vo I can't see anything at all, but I doubt it needs a negative voltage (unless they are unbeliveably sloppy - possible)
I would try setting up an e.g. 1k resistor from Vdd to top of 1k pot, with it's other end to Vss with it's wiper to Vo. IIRC usually the contrast for these LCDs is controlled over a pretty small range around 0 to 1V or so. This setup will give you from 0 to 2.5V, start with wiper to ground and adjust if necessary.

If you're feeling lucky you could just tie Vo to ground (through a smallish resistor if nervous) and see if it works.

In the past with a couple of non critical debugging type setups using any old LCD (usually no datasheet handy) I have just tied the Vo to ground and it worked fine - of course this may not be the case for your particular model but I'd be surprised if it was out of the 0 - 2.5V range.