If there are no moral truths, then serial killers have done nothing wrong for example

This does not follow from moral relativism. Moral relativism simply states the morality of serial killers is determined by people rather than an absolute truth.

For example, if you add the detail of “serial killer of humans”, most societies would deem that morally wrong. In contrast, “serial killer of wasps” would be considered perfectly fine by many. A moral relativist would say the difference between these two is determined by society.

You can, of course, claim that murdering humans is not morally wrong. A moral absolutist might say “you’re wrong because X”, while a moral relativist might say “I don’t agree because X”.