One possible plot could be that Percy has finally had enough of the gods' interference in his life and the lives of his friends. He might yell at them for their unfair treatment, like how they often use demigods as pawns in their games. Maybe he'll demand more respect for the sacrifices demigods make.
Percy is the central figure for sure. The gods, like Hades, might be there too if the story involves issues related to the Underworld. Grover could also play a role. He is Percy's friend and might be there to support Percy when he yells at the gods, or his experiences with the gods could be part of what drives Percy to yell. Chiron, the centaur, could also be a character as he is a mentor to Percy and could have some influence on the situation.
Well, it could be that Percy yells at Nico because he feels Nico is being too reckless in a dangerous mission. Maybe Nico is about to do something that could endanger the whole group, and Percy, being the protective type, loses his temper.
Probably Poseidon, as he is Percy's father and has a vested interest in his son's well - being. Zeus might also be protective in some cases, despite their past differences, as Percy is a hero who often saves the world which benefits Zeus's realm as well.
Well, it could be about Apollo's over - confidence. He might have boasted about his powers or achievements in a way that annoyed Artemis. In the context of the Percy Jackson universe, Artemis is very protective of her domain and those associated with her. If Apollo made a decision regarding something that was under Artemis' purview without consulting her, like interfering in the affairs of the hunters, Artemis would yell at him. Also, Apollo's past mistakes or actions that went against the values of the gods might be brought up by Artemis during her yelling.
She could also become cold and distant. Instead of engaging in a shouting match, she might simply walk away and give Percy the silent treatment. This can be a powerful reaction as it makes Percy realize that his yelling has pushed her away and he has to make an effort to fix things.
Well, without reading the specific fanfiction, it's hard to say exactly who Percy is. But generally, in a 'champion of the gods' scenario, Percy would be a central figure. He might be a hero with unique powers, like being able to control certain elements or having superhuman strength. He could be a character who has to face various challenges and tasks on behalf of the gods, kind of like a chosen one in a fantasy story.
Zeus is a possibility. As the king of the gods, he might see Percy's potential and adopt him to have an ally among the demigods. Another could be Poseidon, since Percy is already his son in the original story. It would be an interesting twist to have Poseidon formally adopt Percy in a more divine way.
The gods might show their protection in various ways. For example, they could directly intervene in battles Percy is in, using their divine powers to shield him from harm. They might also send him signs or omens to guide him away from danger.
Zeus is often one of the main gods showing possessiveness. He sees Percy as a powerful asset or a potential threat, so he wants to keep Percy in check. Poseidon, as Percy's father, also shows possessiveness in a different way. He wants to protect Percy but also use his powers for his own purposes sometimes.