The clue is in the name....Forklift. As has been stated lift trucks are not designed to be used as compactors because you cannot utilise the lowering capability to apply force per se.
It becomes a question of can the waste be reduced in size before it is placed in the container/skip, thereby reducing the need to try and compact it?
I am guessing that the debris is dropped in from the top and there is not the option to load through the front?