Don John is the illegitimate half brother of Don Pedro and filled with bitterness and resentment. He relates to Hero malevolently, using her as a tool with which to try to hurt Don Pedro through his favorite, Claudio. Don John has no evident animosity toward Hero: she simply happens to be available as a way to hurt his enemies. Part of Don John's villainy is his inability to see Hero as fully human.
Don John manipulates events so that it looks to Don Pedro and Claudio like the pure and faithful Hero is having an affair with Borachio. Don John's wicked plan initially seems successful when Claudio rejects and humiliates the hapless Hero, who takes her suffering without much protest. However, Don John is exposed and the play ends happily.
Both Don John and Hero are conventional, undeveloped characters. Don John is a stock villain, bent on mischief making, though his underlying psychology is not explored. Hero is similar to the perfect heroine from a courtly romance. Shakespeare will be more successful using a similar scenario when he turns to write the tragedy Othello, because he will develop his characters in a more compelling way.
There is no actual relationship between Hero and Don John. Hero's father is a good friend of Don Pedro, Don John's half brother. Hero is like a daughter to Don Pedro and he cares for her and her family. Don John hate Don Pedro and wants to get revenge on him for defeating him. He schemes to do so by humiliating Don Pedro's dearest friend, Claudio. Claudio is in love with Hero so she gets caught up in Don John's plans by association. Don John couldn't care less what happens to Hero and she really has no direct ties or communication with Don John. She is simply the tragic victim of his plans to hurt his brother.