Harry's character would likely become more complex. Initially, he might be distrustful of Voldemort's motives but also feel a sense of obligation for being saved. This could lead to him being more cautious and calculating in his actions, as he has to navigate the dangerous waters of being associated with the Dark Lord.
Often, Voldemort starts as this purely evil figure. But when he adopts Harry, he begins to show some unexpected softness. For example, he might start to protect Harry from other threats, which is very different from his usual self.
He could develop a darker side to his personality. Learning from Voldemort, he may become more ruthless in achieving his goals, even if those goals are still ultimately good. For instance, he might be more willing to use unethical means to defeat other dark wizards. But at the same time, he would also be constantly struggling against this new - found darkness within him. He would be in a battle not just with external enemies but also with his own changing self.
At first, Harry would be conflicted. He has always been against Voldemort, but now being forced to be with him would make him question his own beliefs. His character could become more cynical as he is exposed to Voldemort's dark world.
In such a fanfiction, Harry would grow in his understanding of the dark side of magic. Being captured by Voldemort means being exposed to the darkest and most malevolent forms of magic. This would make him more cautious in his own use of magic, but also more knowledgeable about how to counter dark magic. His character would also become more patient. He can't rush into things as he is in a very dangerous situation. He has to bide his time, wait for the right moment to escape or fight back. This new - found patience would be a valuable asset in his future battles against Voldemort. Moreover, he would develop a stronger sense of self - identity. He would be constantly reminded of who he is and what he stands for in the face of Voldemort's attempts to break him.
Well, think about it. Harry's relationship with his friends is ruined. Hermione might try to convince him to come back to the good side, but Harry, feeling so betrayed by his past experiences, would resist. With Voldemort, Harry would be in a very strange position. Voldemort doesn't really trust anyone, but he would use Harry to his advantage. Harry, in turn, might start to see Voldemort as a sort of father figure, which is a very warped view considering Voldemort's nature. And the wizarding world as a whole would view Harry as a fallen hero, and there would be a lot of fear and hatred towards him.
In this fanfiction, the relationship between Harry and Voldemort would be full of contradictions. Voldemort adopts Harry for his own purposes, but Harry has his own will. In the beginning, Harry may be forced to follow Voldemort's orders out of fear. As time passes, he might start to question Voldemort's actions and ideology. Voldemort, seeing Harry's growing independence, could become more controlling. This could lead to a cycle of rebellion and suppression in their relationship. Harry may also find some common ground with Voldemort in their shared loneliness or desire for power, which would add more layers to their complex relationship.
In such a fanfiction, Voldemort would view Harry as his possession. He'd order Harry around, thinking he can use Harry's wolf - like abilities for his own dark deeds. However, Harry would not go down without a fight. He'd constantly be trying to undermine Voldemort from within. Their relationship would be a very strange and twisted one. Harry would have to pretend to be loyal while actually hating Voldemort with every fiber of his being. And Voldemort, being so arrogant, might not realize how much of a threat Harry still is even in his wolf form until it's too late.
Harry would likely become more cunning. He has to survive in Voldemort's lair, so he'd learn to hide his true feelings and intentions. For example, he might pretend to be more submissive than he really is to gain Voldemort's trust slightly.
The relationship would become extremely complex. Harry would be confused and conflicted. He's been taught all his life that Voldemort is pure evil, but now he's been saved by him. Voldemort, on the other hand, might start to see Harry in a different light, perhaps as a tool or as someone with whom he can form an unusual bond. It would be a relationship full of mistrust on Harry's part and manipulative intentions on Voldemort's.
One plot idea could be that Voldemort saves Harry from his abusers in the Dursley household because he realizes Harry's potential is being squandered there. He then takes Harry under his wing, not to turn him evil, but to train him in magic in a very different way than Hogwarts. Maybe he wants to use Harry as a tool against the Ministry, which he believes is corrupt.