The 3rd Bleach Character shows a bit of Ulquiorra's past in it.

Not a huge amount of detail but it does shed light on one thing. After Ichigo and Ulquiorra fought, Ulquiorra started to turn to dust. He then told Ichigo to finish it by killing him with his sword, otherwise 'it would never end'. As someone pointed out in the comments this is likely referring to the 'Void' Ulquiorra was talking about. Presumably Ulquiorra did not want to return to such a place as it was literally 'nothingness'