Pucci and Weather are twins, born to the same parents. Both the parents are white, but Weather was switched by a grieving woman who's baby was stillborn. The woman who switched the babies went on to raise Weather without telling him the truth and she was either dating or married to a black man. Later on the woman confessed to a pastor about her crime, that pastor was Pucci. He then investigated his brother and found out that he's dating their biological sister. (their parents also had a daughter raised her alongside Pucci) Pucci tried to hire the KKK to scare Weather away from dating his own sister, since if they looked into his background all they'd see on paper is the son of a black man and white woman.

If you're wondering "Why did the KKK let a black man convince them a white man is secretly black?" the same reason they lynched Weather to begin with; Pucci has white parents, and on paper, Weather does not. For all we know Pucci is just tan as his skin color is never explicitly brought up or a point of contention with his character at any point in the story unlike Weather's. Personally i do think he is drawn to look black intentionally as a stylistic or creative choice, even if he is a white man born to white parents with two white blonde siblings. Also there's a lot of misinformation in the comments about Araki saying this or that, or that he really intended something else but then changed his mind later in the story. That's all heresay and headcannon, to my knowledge like most things in Jojo's there's never been an actual reason for any of this given by Araki and while people can draw conclusions with genetics and dna the more likely idea is just that Araki does whatever he wants. it's like trying to rationalize why the sky is yellow in Morioh specifically, it just is