How can one atone for showing private pictures of a (deceased) friend to a relative, after sincere repentance and deep remorse?

You erred by showing your relative the woman's picture, as it is assisting in a forbidden act. Sheikh Ibn Baz issued a fatwa prohibiting looking at pictures of women due to the temptation they cause, based on the generality of Allah’s saying: "Tell the believing men to reduce [some] of their vision and guard their private parts. That is purer for them. Indeed, Allah is Acquainted with what they do." What is incumbent upon you now is to repent to Allah Almighty and to increase in righteous deeds, for Allah Almighty says: "But whoever repents after his wrongdoing and reforms, indeed, Allah will turn to him in forgiveness. Indeed, Allah is Forgiving and Merciful." And increase in supplication and seeking forgiveness for your colleague, and strive for the guidance of your relative.