One characteristic could be extreme and unhinged interpretations of the characters. For example, making Harry Potter into a truly deranged and unpredictable figure, far from his original heroic self. It might also involve very dark and twisted plotlines that go against the general spirit of the original series, like having characters engage in immoral acts that were never part of the canonical story.
Well, often in 'psychotic harry potter fanfiction', the settings can be distorted in a very abnormal way. Instead of the relatively positive and magical Hogwarts we know, it could be presented as a place full of horror and chaos. The relationships between characters might also be warped, such as turning friends into bitter enemies in the most cruel and illogical ways. And the magic itself could be used in disturbing and malevolent ways rather than for good as in the original books.
In 'psychotic harry potter fanfiction', there may be a blurring of the line between good and evil in a very extreme sense. The story might make the good characters question their own morality in ways that are not at all in line with the original story's values. The narrative could also introduce new elements that are completely out of the blue and have no connection to the established Potterverse, like some sort of alien - like magic that is not part of the traditional magic system. This can create a sense of unease and 'psychosis' in the overall feel of the fanfiction.