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.
I'm not sure exactly which 'guts' you mean by 'guts back story'. However, if it's in a general sense of someone having the 'guts' (courage) and their back story related to how they developed that courage, it could be that they faced a lot of difficult situations in life. For example, they might have had to stand up to bullies when they were young, and through those experiences, they gradually built up their inner strength and 'guts'.