I'll admit the first time I saw Chrom while playing through the demo of Fire Emblem: Awakening I wasn't entirely sure what to think about him. Mostly I think it was just his voice actor as I thought the voice was too mature for the character; I thought he looked interesting otherwise. By the time the actual game was released, however, that was it. I liked Chrom even more, and even considered him among my favorites, but I don't think it was until much later that he became one of my absolute favorite characters, topping perhaps even Ike and Ephraim who were already my favorite male characters in the series. I couldn't tell you when that was or how that happened though.
One reason why I like Chrom so much is he's not your typical royal. He's straightforward and honest, and treats everyone with respect, regardless of who they are or where they came from. While stuffy royals can be quite entertaining, they can't hold a candle to those who wear their hearts on their sleeves and care strongly for their people. I just find them to be so much more likeable because of this, and Chrom is no exception. Although the Fire Emblem series is full of nobles/royals like this––Eliwood from Fire Emblem comes to mind––Chrom stands out to me the most. I think the reason for that is because he actually goes out of his way to help the people, and doesn't spend all his time in the castle (and if he is in the castle he's training so that he might better protect his people). While formal talk can be cool, Chrom's direct manner of speech is refreshing.
The transformation Chrom goes through throughout the course of the game is also pretty incredible. He goes from a brash young man, always running head first into a fight, to a beloved leader, much like his sister before him. One of my favorite moments which show this is with Gangrel. When we first meet him in Chapter 5: The Exalt and the King Chrom wastes no time to fight his forces when he threatens to hurt Emmeryn. When we meet Gangrel again in Paralogue 18: The Dead King's Lament Chrom actually tries to recruit him. Chrom claims that while killing Gangrel would give him great pleasure, it would put him no closer towards his goal to defeat Grima. If that's not a sign of transformation for Chrom then I don't know what is.