What is the appropriate stance to take towards a friend who, through the questioner's actions, caused harm by entering into an unlawful relationship with his sister, and is this considered an injustice and a violation of people's rights, and what is the way to forgive him and mend the relationship?

Establishing an illicit relationship with a foreign woman is an evil deed in Islamic law, and the fact that she is your friend's sister makes the matter worse, as friendship entails safeguarding honor, not violating it. It is from faith to feel protective over the sacred limits set by Allah. Your regret over the sin is valid, but you must initiate sincere and complete repentance, which includes other conditions besides regret. As for the right of the woman's guardians over you due to the violation of their honor, we hope that sincere repentance and an abundance of good deeds will expiate it, given the difficulty of seeking direct forgiveness in matters of honor due to the anger it might provoke.