One challenge could be controlling the assassin's violent tendencies. Since the person is being groomed to be an assassin, it's likely they have a natural inclination towards violence, and Harry has to find a way to temper that while also teaching self - defense.
The assassin's loyalty could also be a big challenge. Harry has to ensure that the person he's raising uses their skills for good and not turn on him or the wizarding world. There might be constant temptations for the assassin to use their abilities for personal gain or be swayed by dark forces, and Harry has to find ways to keep their loyalty and moral compass in check.
It could be that the assassin was initially a victim of some dark wizard's experiment. Harry, being the hero he is, feels responsible for the child and decides to raise him. In the process of raising, he unknowingly nurtures the assassin-like qualities in the child because of the unique circumstances and challenges they face together.
He might face political intrigue. There are likely those in the Wizengamot who oppose him due to his past actions and his association with Dumbledore. Also, he has to deal with the complex and often archaic laws and traditions of the wizarding world which could be a huge obstacle.
In the fanfic, the main challenges for Harry could be related to the limited time he has. He has his own studies and tasks to attend to, like preparing for exams or dealing with the mysteries of the Chamber of Secrets. Yet, he still has to find time to protect Colin. Also, the magical environment itself can pose challenges. There could be spells that are difficult to counteract while protecting Colin, especially if they are new or dark magic.
One challenge could be the basilisk's own will. Even if Harry has the ability to control it, the basilisk is a powerful and ancient creature with its own instincts. It might resist Harry's control from time to time, especially when it comes to situations that go against its natural behavior, like being made to not attack innocent students.
The Dementors are a huge challenge. Their presence alone can drain his happiness and will to escape. He has to find a way to avoid them or protect himself from their effects.
One challenge could be the unfamiliarity with the Fairy Tail world. Different rules of magic and different types of magical creatures might be new to Harry. Another is that the Fairy Tail characters themselves might be very powerful and have their own unique abilities that Harry has to figure out how to counter.
He also has to deal with the secrecy around wand - making. Since it's a highly guarded art in the wizarding world, he might face opposition from the Ministry of Magic or other wizards who think he's meddling in something he shouldn't. Plus, he has to make sure that the wand he crafts is unique to him and suits his style of magic, which is no easy feat.
One challenge could be dealing with the traditionalist teachers and pure - blood families. They might resist any changes Harry tries to make as they are used to the old ways. For example, if Harry wants to introduce muggle - born friendly policies, some of the old - school pure - bloods might oppose it.
One challenge could be betrayal. There might be some so - called allies who are secretly working for Voldemort. Another is the lack of resources. If he is constantly on the run, he may not have access to enough food, water or magical supplies. And he has to be constantly vigilant against dark magic spells that Voldemort might send his way.
Obviously, Harry and Charlus Potter are the main characters. Harry as the one being raised, and Charlus as the one doing the raising.