I've talked about this briefly, but never in depth.
But this brings up a nature v. nurture debate in me. I do think a lot of Hope is just who she is. But we can't deny how her life shapes her.
The first time in canon we ever get a sense of Hope's personality is when we meet her as a seven year old, who is being raised by Hayley alone. Hope is a gentle, sensitive, and caring girl. I do think that is the baseline for Hope.
However, if she had been raised with all of the Mikaelsons, I can't help but feel that she would be corrupted slightly. We met Marcel briefly as a child and he was also a sweet and caring child. However, the Mikaelsons made him like them. Marcel was still caring, but there was an edge to it. He was willing to starve and imprison his best friend for betraying him. He was willing to lock Davina, a child, up for his own gain. He was willing to unleash a horde of vampires on New Orleans to defeat his enemies (the Mikaelsons). He was willing to hurt those he claimed to love for power. Marcel became a Mikaelson through and through. The same thing would have happened to Hope.
Legacies, from what I've seen, loves to talk about Hope being Klaus' daughter, but in reality, canon Hope is Freya and Hayley's daughter. And not even the Freya we first met, but the Freya they turned her into after they killed Cami off. Being raised by them and feeling abandoned by the rest of her family really shaped who she became. For better or worse.
I also have trouble with this question because it depends what version of Klaus stuck around. We saw various different versions of Klaus as a father, but one thing was consistent is that he wasn't a very good father. He wanted to be, but let his fear and quest for power interfere. Even when he was around for Hope's life, we see him often handing Hope off to Freya or Elijah or Jackson and Hayley or just vaguely to the bayou.
But, for arguments sake, let's say he was actively invovled in raising Hope. He would spoil her and teach Hope that the world owed her everything. We would have seen a much more entitled Hope. She would still be caring underneath, but like Marcel would feel that she was owed more than she was given and do questionable things to get there.
I do think Elijah would try to temper this more, he saw what his hands off approach did to Kol, Klaus, and Rebekah. Elijah and Hayley would have had to be the disciplinarians, constantly fighting Klaus and his inability to actually parent. I would hope that we would still see Elijah realizing that the pressure he put on Hope as a baby was too much and he would be able to tell her that earlier than he did in canon.
We also wouldn't see Hope feeling the pressure the way she did because their wouldn't be the sacrifices. Hope's point system she kept was because she felt like she had to make the Mikaelsons' sacrifice worth it. That she had to be worth it. No sacrifice, no pressure. Even if Elijah didn't realize his mistake, there would be nothing telling Hope that it was her job to "save" Klaus.
This is good, but also would just feed into the entitlement with nothing to temper her out. Rebekah and Kol would also spoil her, encouraging her problematic behavior. Freya, even in canon, often viewed Hope as an adult rather than a child, letting her make decisions that she was too young to be making.
But the saving graces would be the in-laws and Hayley. We see Hayley really mature as a mother. Part of this is likely because she was doing it on her own, but I think she would still have this growth with the Mikaelsons, encouraging them to have their own growth. Cami would have forced Klaus to confront his issues that made him a questionable father and showed him how problematic his techniques of parenting were. Davina would have been one of Hope's biggest advocates, knowing how much pressure was on Hope. Same with Marcel. He knew how much the Mikaelsons broke him and he wouldn't let them do that to Hope. Keelin would also have pointed out all of their problematic behavior, never letting them get away with it.
Hope would have been similar to the Rebekah we met in TVD. Kind of a bratty teenager who has a good heart underneath it all. She would grow and mature and hopefully find her own footing outside of the Mikaelsons. She would also be less afraid of the "darkness" inside of her because there wouldn't be the lingering ghost of the Mikaelsons hanging over her. They would just be her family. Instead of learning of them from books and people who hated them, she would be able to form her own opinions. She would learn to embrace all parts of herself as she got older.