I think you can drop the 8255 from the picture if this is your first attempt with the LCD. And particularly so if you are not conversant with the 8255 and its various modes. Trying to get each one going ( 8255, LCD) is quite job for the first time user and trying both together is sure asking for trouble.

I am not sure if you are so hard up for port pins - at a very basic level to drive the 20x2 LCD you just need the following :

4 pins for the data.
1 pin for register select.
1 pin for Enable
1 pin for R!W ( if you only want to write and not read the LCD registers then this also can be done away with )

Just make sure that the data is set up first, then the RS and R!W and then the Enable in that order. The LCD will then work provided it is wired properly / not damaged / contrast correctly adjusted / properly intialized.

Defenitley not easy the first time - but once you get that first character showing, it is child's play thereafter.