Having seen the movie,
Arthur's mom had been traumatized in her younger days, while working for Thomas Wayne. She allowed her then boyfriend to abuse her adopted son-this got her committed, and in the newspaper. There was never an affair between them. Arthur believed it to be true because his adopted mother-who he believes at this point is his real mother(having just found her note)-told him so. The fact that Thomas is rich and that he could have used his money to cover it up(the current state of the city suggests this is possible. this is proven false due to physical evidence of the abuse****head trauma and burns on Arthur's back), the viewer POSSIBLY believes they had an affair.
The truth is, she projected her reality as real reality, and wrote a note to who she still believed was Arthur's father(she believes he his her real son). In spite of the history, she still looks up to Thomas as a good man-because in truth, he did care for his employees. He even helped keep things easy on Arthur by not calling the police at Wayne Manor(knowing full well of her accusation of him being Arthur's father). In the scene where Arthur finally realizes his true past, you hear about the symptoms of her mental disease(projection being one of them).
No blood relation, but a connection.
Joker hates Wayne and those who look down on every little person(even if they are really good), and Bruce will hate that the Joker indirectly helped cause the death of his parents.