Best Battery pack solution for raspberry pi and lcd
I'm thinking about using a battery pack of 6 AA (12V)


An AA-battery is usually 1.5V, so 6 AA-batteries connected in serial would be 9V and not 12V. Furthermore, depending on the specific AA-batteries, they might not be able to deliver the 4A that your screen requires and the current that the rest of your system requires.


  My lcd screen requires 12v and 4A


Just make sure that the LCD display really requires 4A. It seems unlikely to require so much current. Consider using an LCD which requires less current if it is the case.

I would suggest using a Lithium Ion (LiPo) battery. They are commonly available at hobby shops and on-line. Usually, they can output high amounts of current. Be sure to design some protection into your circuit against over-voltage, over-current, accidental short circuits, polarity protection etc. 


  is it possible to charge them


LiPo battery chargers are commonly available.


  From what I understand so far, i need to wire the battery to a regulator


You are correct that you can use a voltage regulator to step down the 12V voltage to 5V for your Raspberry Pi. However, a linear voltage regulator like the one you suggest in your question dissipates power and will become hot over an extended period of time. The difference between the 12V and the 5V output voltage for the voltage regulator is also a relatively large voltage drop which will cause the heat generated by the voltage regulator to increase. Consider putting a heat sink on your voltage regulator.

A better option would be to use a buck converter for the Rpi, or even better get a battery with a higher voltage (14.8V, 4S LiPo Battery) and use two buck converters - one for the Rpi and one for the LCD. A buck converter is more efficient than a linear voltage regulator.

Edit:

As someone in the comments mentioned, rather find an off-the-shelf solution in terms of protections circuitry. There can be a fire hazard with LiPo batteries if you don't know what you're doing.