The scenes between Charles, The Holy Roman Emperor, and Catherine of Aragon. You can tell that he truly cares for his aunt, and her happiness. There is a genuineness to his concern for her that goes beyond the political, when he is told that Henry is tiring of her.
Most scenes between Mary and Elizabeth. Specific ones include Mary comforting baby Elizabeth in season 2 (extra points since Mary has every reason not to care if this particular baby is upset), the two sisters snuggled in bed and talking about their new brother in season 3, and Mary telling Elizabeth that they've both been restored to the succession in season 4.
Any scenes between Katherine of Aragon and Mary or Anne Boleyn and Elizabeth.
Any scenes where Henry shows affection for his children.
The last meeting between Henry and Charles Brandon.
Many of the scenes between Mary and Chapuys, especially in the fourth season.
At the end of the show, Henry sees Anne Boleyn (either as a ghost or his own imagination) and she tells him that although she desperately wanted to give him a son, she is now so proud of Elizabeth.
At Christmas in Season 3, when Henry is presiding over the festivities with his new Queen, Jane Seymour, and his newly-reinstated daughter Princess Mary. Mary and/or Jane have arranged to bring his most-recently-delegtimized child, Princess Elizabeth, to court, and Henry publicly calls her his daughter (important as he'd had her mother executed for adultery and speculated Elizabeth wasn't his) and lifts her up to sit beside them. He announces to the court, "Je suis en familie (I am among my family)."
The procession of the King as he journeys north, with all the people cheering the royal family.