Perhaps Percy was kicked out because of a prophecy that the camp elders misinterpreted. They thought that Percy's presence would bring great danger to the camp, so they made the hasty decision to expel him. It could also be that there was some sort of political intrigue within the camp. Some powerful figures at the camp might have seen Percy as a threat to their power and influence, and so they fabricated reasons to get him kicked out.
One reason could be if he was wrongly accused of stealing something important from the camp. Maybe a powerful magical item goes missing and Percy is the prime suspect because of his past issues with authority. Another reason might be if he accidentally endangers the camp in some way during a mission or training exercise. For example, if he unknowingly leads some dangerous monsters back to the camp.
He might be really angry at first. He's given so much to the camp, and being kicked out like that without a proper explanation would make him furious. He could storm off, vowing to find out the truth and clear his name.
One reason could be that he feels a sense of being out of place. Maybe new campers have come in and he no longer feels as important or needed. He might also leave in search of a greater purpose. In some fanfictions, he could be on a quest to find something that he believes is crucial for the safety of the camp, but he has to leave the camp itself to do it.
A plot could be that Percy accidentally breaks an ancient and sacred rule of the camp while trying to save a fellow camper. As a result, he's kicked out. He then decides to seek refuge with his mortal family, but soon realizes that monsters are targeting them too because of his connection. So he has to protect his family while also finding a way to get back into the good graces of the camp.
One possible reason for Percy being kicked out in fanfiction could be that he was wrongly accused of stealing a powerful artifact from the camp. Maybe someone framed him to get him out of the way. As for what happens next, he could go on a journey to clear his name, finding allies outside the camp who believe in his innocence.
Percy could also be very sad. He has so many memories and relationships at the camp. He might go into a state of depression for a while, not knowing where to go or what to do next. But then, he would probably start to think about how to get back in or how to make things right. He might start looking for evidence to clear his name if he was wrongly accused. He could also look for new allies outside of the camp who could help him in his situation.
One reason could be to avoid a dangerous prophecy that they believe the camp can't protect them from. In some fanfictions, they might think that by running away, they can face the threat on their own terms.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
He might leave because of a personal conflict. Perhaps he has a falling - out with some of the other campers or with the camp leaders. Like if he disagrees with a new rule or policy that the camp has implemented which he thinks is unjust.
One possible reason could be a dark turn in his character. Maybe he's been corrupted by some external force like a powerful and evil magic. Another reason might be a misunderstanding. For example, he believes the camp has betrayed him in some way, so he lashes out in anger.