Postmodern Philosophy and Morality Simone de Beauvior and Jean Paul Sartre had an existentialist view very similar/ influential to postmodernism. They believed, (or at least Sartre did) the opposite of the conventional view that postmodernism (in their case, existentialism) reduced moral responsibility. Sartre holds that each person is maximally free, since existence precedes essence, we are completely free to shape our life however we choose. Subscribing to a pre determinded moral system would probably constitute "bad faith" (not 100% sure I'm using bad faith correctly here) since in doing so, we attempt to transfer the responsibility of our actions to a pre-existing structure, making it so that we don't have to engage in the difficult, but fulfilling task of sincerely asking ourselves, questions like "what do I want?" "what should I want?" "What can I do?" "What should I do?", and coming up with detailed answers that are relevant to our lives. Though I don't know her position on it, de Beauvoir writes a whole book on this subject called The Ethics of Ambiguity.
More generally, the postmodernist view on morality (the version I would defend) in fact transcends the binary good and evil, quite simply by observing the historical development, and therefore the severe imperfections of moral systems. There is no such thing as morality, instead there are moralities, and it is ultimately up to each individual person to make personal judgments as to which moral systems they should subscribe to, if any.
A clear illustration of this, is that growing up, I was raised in a Catholic household and I played Ice Hockey, before I would go to a game, my mother would tell me that I shouldn't hurt anyone on the ice, by being too rough and playing physically. At the rink, my coach would tell me that I shouldn't let the team down by being too soft and not playing physically. Both of these "shouldn't"s were in the moral sense, and the conflicting sentiments at play can pretty clearly be traced historically to what Nietzsche calls Christian, altruistic, slave morality, and Pagan, assertive/ ability based, master morality.
Which moral system should I follow? Ultimately I had to make my own judgment, because any appeal to morality would be entirely contingent on which moral system I independently decided I preferred. To oversimplify, this is where power dynamics become incredibly important because since there is no perfect universality for any one moral system, the moral system which has the most sway is decided by its cultural prevalence. This is (partly) why holy wars, inquisitions, educational projects, laws, prisons, social stigmas, and other means of propagating and enforcing moral standards are typically used, rather than some objective appeal to the "logical merit" or what have you of moral systems, at least as it applies to humanity's practical, day to day use of moral behavior.
Another thing Nietzsche would have pointed out is the inherit danger of morality. When (and usually only when) you understand another person to be evil, it becomes okay to hurt them. Ideally, evil people "deserve" to be hurt, but in practice, all you need is some really good rhetoric or some blatant lies to turn people against each other, with the powerful (and very dangerous) emotional vindictiveness that comes using with moral language. In the 201st aphorism of Beyond Good and Evil Nietzsche says:
"Is it not sufficient if the criminal be rendered HARMLESS? Why should we still punish? Punishment itself is terrible!"--with these questions gregarious morality, the morality of fear, draws its ultimate conclusion. If one could at all do away with danger, the cause of fear, one would have done away with this morality at the same time."