As a brother.
As the middle child of three siblings, Chrom has an older sister and a younger sister. Emmeryn being the oldest and Lissa being the youngest. The three are quite close and have likely always been that way, even when they were young. Sadly the game doesn't seem to go into much depth in his relationship with Lissa, and instead seems more focused on his relationship with Emmeryn. As such I am just going to focus on Chrom and Emmeryn here. I might change my mind on this in the future.
Fifteen years ago, when the three of them were still young children and their father was the exalt, Ylisse and Plegia were embroiled in a war which lasted for many years, ending only with his death. Shortly before Emmeryn's tenth year, she was crowned the new exalt of Ylisse. Although the people seem to love her when we first see her at the end of Chapter 1: Unwelcome Change, this was not always the case; it took many years and a lot of pain on her part to heal her people––only Chrom and Lissa understood the pain she was going through. Emmeryn was the target of hate from the people, and even her own subjects would hurt insults at her. The people mocked and vilified her. Despite these things, Emmeryn never once resented them for it, desiring only to reach out and heal their pain. According to Chrom, Emmeryn represents the best part of the Halidom––the part most worth protecting. She is peace.
The relationship Emmeryn and Chrom share is truly incredible. When it comes to Emmeryn, Chrom is the protective younger brother who cares strongly for her well being, and he isn't afraid to show it. Anyone who threatens her soon finds themselves on the sharp end of his sword. However once she has made up her mind on something he will not argue with her final decision, regardless of how against the idea he may be. Sometimes this selflessness of hers bothers Chrom and he asks her to be selfish for once in her life. Emmeryn always smiles and tells Chrom and Lissa how much she loves them. To her peace and the happiness of her people are more important than what happens to her. I think also though that the happiness of Chrom and Lissa are as equally important to her, and that the peace she desires is for them too. At least I think so. She doesn't outright say it.
"I love you, Chrom. Both you and Lissa are my everything. As for the peace I seek... You cannot see who it is for."
– Chapter 7: Incursion (Emmeryn to Chrom)
When Emmeryn is captured and sentenced to death by the Mad King of Plegia, Chrom wastes no time in mounting a rescue attempt. When their victory seemed guaranteed the unthinkable happens and Chrom is forced to make a choice––hand over the Fire Emblem, Ylisse's greatest treasure, or watch Emmeryn die. Regardless of what he decides to do Emmeryn makes the choice for him so that he doesn't have to choose. She shares a few words with the people of Plegia and then falls to her death. Although Chrom rushes in an attempt to save her he is too late. Chrom is devastated by her death but he does not drown in despair or let her sacrifice be in vain, and instead works toward creating the sort of world Emmeryn wished to see and bring peace not only to Ylisse, but to all the realms.