T'Challa and T'Chaka only share one scene prior to T'Chaka's death but they establish that T'Chaka is proud of his son and sees in him a strong, future king for Wakanda. T'Challa too greatly respects his father.
T'Challa didn't really interact with Stark that much. His agenda is focused entirely on Bucky and he doesn't mingle with any of the superheroes outside of some minor dialog with Black Widow. It's best summed up when during the big airport battle Hawkeye is like 'I don't think we've been introduced, I'm Clint!' and T'Challa just responds with 'I don't care'. Again, he only cares about getting Bucky.
There's only one Dora Milaje shown as his bodyguard when he leaves some government compound. She's like a bald Grace Jones, although better looking and has a goddamn nice booty. Black Widow stands in the way because she wishes to speak with him and she just directly threatens her to get out of the way. T'Challa smirks and says that although he'd love to see that fight, this is not the time and stops her.
I'd say it's roughly equal time between T'Challa in and out of the suit, and Boseman handles the accent fine. It's consistent and it sounds genuine.
There's three fights I'd say. First when he first locates Bucky which leads to the big tunnel chase, where he prevents his escape fighting off Bucky, Cap and Falcon individually. Later when a brainwashed Bucky escapes from custody he fights him again, and actually bests him in direct combat although Bucky gets away because he falls of a ledge and by the time T'Challa catches up Bucky's gone. Then he shows up for the big airport fight midway in the movie, where he primarily faces off against Cap (goes toe-to-toe with him for a good while) and dodges every arrow Hawkeye fires at him, as well as catching two explosive arrows which he shrugs off. The closest loss he takes is getting swept aside by Giant Man, and Black Widow zaps him repeatedly to slow him down from catching up to Steve and Bucky, although he doesn't go down. Since there's no real clear winner in any of the battles he (nor anyone else) really wins, but it's made clear that in combat he could best the most of them easily.
The stolen Vibranium from AoU is mentioned by king T'Chaka at the UN assembly, and he says it is the primary reason Wakanda has stepped away from its isolationist history, because they feel partly responsible for what happened and felt the need to start reaching out to other nations.
T'Challa doesn't bring up Vibranium but Cap does ask him whether his suit is made of Vibranium, and T'Challa just remains silent.
As said, T'Challa gets the most dialog with Black Widow, mostly because she directly apologized to him for the Wakandan lives lost in the opening scene of the movie. He also has a strong moment towards the end when he finds Zemo, the real killer of T'Chaka, and upon seeing how driven by revenge he is, and how revenge is tearing the Avengers apart, he doesn't want revenge to eat him up so he puts away his claws and vows judgment will be served. Zemo then tries to kill himself but T'Challa stops him and chokes him out saying 'The living aren't done with you yet'.
Bucky is being treated in a high-tech Wakandan lab. T'Challa joins Cap as they discuss the future of Wakanda if the world finds out they're hiding Bucky, and T'Challa responds that he loves to see them try. The camera then pans away to the outside where it's revealed the lab's in the middle of a thick jungle overseeing a giant panther statue. That's also the closest we get to any idea for what might go down in the Black Panther solo flick.