It could be some of the other trainees who are jealous of Percy's progress and his special relationship with Chaos.
Perhaps there are beings from the chaotic realm who oppose Chaos' decision to train Percy. They might see Percy as an outsider and a threat to their way of life in that realm. These antagonists could try to sabotage Percy's training in various ways, like setting up dangerous traps during his training exercises or spreading false rumors about him to turn other trainees against him.
Maybe there are some dark forces at work. They sense Percy's growing power and want to stop him before he becomes too strong. These could be ancient evils that have been awakened by Percy's power development. They could send minions or create illusions to make Percy's training difficult.
Well, perhaps Percy could be learning some new and powerful combat skills from Chaos. Maybe Chaos teaches him to manipulate chaos energy in unique ways for battle.
The Titans could still be a major threat. Since Percy is now the Host of Chaos, they might see him as a new opportunity to regain power. Kronos could try to manipulate Percy's chaotic powers for his own gain, or maybe use Percy's new status to cause more chaos in the world and then step in to 'restore order' in a way that benefits the Titans.
The Titans could be great antagonists. They might see Percy as a threat to their plans of overthrowing the gods again. Since Percy has chaos powers, they could fear that he could disrupt their carefully laid - out schemes. Another possibility is some of the more traditional enemies of the demigods, like the monsters. They might be drawn to Percy's chaotic energy and try to either consume it or destroy him to prevent him from becoming too powerful.
Maybe the gods or powerful beings who are trying to maintain the current balance and see Percy's new role as a disruption. They would try to stop him from carrying out his assassinations in the chaos world.
The Akatsuki could be antagonists. They might see Sakura being trained by Madara as a threat to their plans. Since Madara has his own motives and is associated with the Akatsuki, they could fear that Sakura will become a powerful force against them if she is fully trained by him.
It could be the agents of Order who see Percy and Thalia joining Chaos as a threat to the balance they strive to maintain. They might send their best warriors to stop Percy and Thalia from carrying out Chaos' plans.
It could be the entities that thrive on the disorder in the chaos. They might see Percy and Bianca as threats to their way of life and try to stop them from achieving whatever goal they have in the chaos.
The traditional village leaders who are set in their ways could be antagonists. They might see Naruto's chaos - inducing actions as a threat to their power and order.
A wise old shaman from an Alaskan tribe might be a supporting character. The shaman could have deep knowledge of the spiritual side of Alaska and guide Percy in his training, perhaps helping him to connect with the forces of nature there.