The reason for going on a killing spree in the first place. From what can be gathered, the Postal Dude believes he's fighting against some kind of great evil that threatens humanity. Of course, given his diary entries and sadistic enjoyment of causing death and destruction, he's probably just flown over the cuckoo's nest.
The diary entries in between levels, e.g. "10/17/97 The Earth is hungry. It's heart throbs and demands cleansing. The Earth is also thirsty..."
The screams of the dying seem to qualify, as do the unsettling menu screens, very unsettling loading screens, and the total Body Horror of the credits screen.
Not so much the original game, but Apocalypse Weekend does introduce undead zombies into the mix, as well as many creepy encounters with zombified Gary Colemans (probably rendered more creepy after his death). A fanmade game, Eternal Damnation, included in the Fudge Pack edition of Postal 2, is full of Nightmare Fuel, especially in parts of the game where giant Body Worlds-like corpses chase you down the street.
In one house, you find that the bathroom floor, sink, and bathtub are covered in blood.