As a father.
After the Mad King Gangrel's defeat the war with Plegia was ended. Although it would take much to heal the wounds of Ylisse, as the war had taken a heavy toll on the halidom, peace would return. Out of respect for his late sister Emmeryn, Chrom had foresworn the title of exalt, but even so the people were eager for the prince to marry. In his heart he knew who that would be. Two years passed and during that time Chrom and his wife had a daughter. Her name was Lucina.
Lucina was an adorable baby who would grow up to become a strong and beautiful woman. However, if the future was not changed she would lose both her parents and the fate of the world left uncertain. That is why Lucina traveled to the past; to change events so that her dark future would not come to pass. Choosing to keep her true identity a secret, she wore a mask and assumed the identity of "Marth", and worked to alter certain events––events that would directly lead to the death of her father and the resurrection of the fell dragon Grima. While she was able to change things, she was not able to prevent them from happening entirely, such as Emmeryn's death. While Lucina was able to prevent her assassination, and the theft of the Fire Emblem, Emmeryn was still fated to die, albeit through a different way.
At the end of Chapter 13: Of Sacred Blood, things seem safe after Chrom and company fight off a Risen attack on the Carrion Isle. However, a single Risen assassin appears and threatens Chrom's life. Out of nowhere Lucina appears and rushes to his side to aid him. With a slip of tongue she calls him "Father" as she deflects the attack. Having failed to kill the prince the Risen retreats and Lucina is relieved. However, a new problem has arisen. Realizing what she had just said she asks that they speak in private. She decided her only option is to tell him everything––about who she is, about her mission, and about the future. Telling him only to look into her eye he could see the Brand of the Exalt on her left eye, the same mark found on the same eye on his child. Right away Chrom knew who she was; she was his daughter. Apologizing to her she then burst into tears, crying in her father's arms.
After that Lucina decides her best course of action is to travel with Chrom. The two are then rarely seen apart. As they spend more time together Lucina learns more about the man her father was, or in this case is, and realizes that if he were to die the world would be robbed of a great man. That's why when she learned that it was Robin who was responsible for killing her father in her own time she thought that killing Robin would be the best way to prevent Chrom's death. Although his reaction can vary depending on their relationship the end result is the same; Chrom steps in and convinces Lucina that their bonds are stronger than ever, and together they can change things. She then trusts them to prove her wrong and that the future would fall to pieces before their bonds do.