First, start with his past. Show how his experiences as the Winter Soldier have made him who he is at the start of the fanfic. Then, gradually introduce the influence of the reader's daughter. Maybe she shows him kindness, and that starts to chip away at his cold exterior. As the story progresses, he should start to question his old ways and beliefs. By the end, he could have made a significant change, like becoming more open and less of a weapon, more of a protector with a new sense of purpose.
Guts is purely fictional. It's a concept or character that exists within the realm of imagination and creative storytelling rather than being based on reality.
Perhaps in this fanfic, Griffith has turned into something truly evil and a threat to the world Guts cares about. Guts, being the warrior he is, might feel it's his duty to put an end to Griffith to save others. Also, Guts could be driven by his own sense of justice. If Griffith has wronged innocent people, Guts may decide that killing him is the only way to set things right.
It depends on the specific fanfic. Some fanfics might have Guts killing Griffith as a major plot point, while others could take a different direction and not have that happen at all.