Joker is chaotic and unpredictable. He thrives on disorder and finds joy in causing mayhem. Bruce Wayne, on the other hand, is all about order and justice. He's constantly trying to keep Gotham in line. So when they meet, Joker's wild and unhinged behavior directly clashes with Bruce's calm and calculated nature. Joker might do something completely random just to see Bruce's reaction, while Bruce is trying to analyze Joker's motives to stop him.
One possible plot could be that Joker, in his twisted way, tries to make Bruce Wayne see the 'fun' in chaos. He might set up elaborate pranks around Gotham that target Wayne Enterprises, forcing Bruce to confront him not as Batman but as Bruce. Another plot could involve Joker finding out Bruce's identity in a unique way and using that knowledge to toy with him psychologically, perhaps by threatening the people Bruce cares about in his civilian life.
Definitely not. In comic books, Bruce Wayne is Batman and the Joker is his arch-nemesis. They have no familial connection. Their dynamic is based on good versus evil, not kinship.