Well, it could start with Harry using his influence within Hogwarts and among his friends. They all start an investigation into Sirius' case. They find out that the Ministry was hasty in their judgment due to Voldemort's influence at the time. With this new understanding, they demand a trial for Sirius. Harry could also use his connection to Dumbledore to help push for the trial.
In such a fanfiction, perhaps Harry discovers some new evidence that was overlooked. He might find a witness who was too afraid to come forward before. Then he presents this evidence to the Ministry, forcing them to give Sirius a trial.
One possible plot is that Harry stumbles upon some old Ministry records that prove Sirius' innocence. He teams up with Hermione, who is brilliant at research. Together, they figure out that Sirius was framed by Peter Pettigrew in a much more elaborate way than they initially thought. They then gather all the evidence and present it to the Wizengamot. With the help of some sympathetic members, like Professor McGonagall, they are able to get Sirius a trial. Harry has to fight against the old prejudices within the Ministry and the fear of Sirius being a supposed mass - murderer. But with his determination and the truth on their side, they manage to get the trial started.
The fanfiction could start with Sirius being magically transported to a new, more just courtroom. Dumbledore could be secretly working behind the scenes to ensure a fair trial. The trial could be full of twists and turns. For example, the dementors that were involved in his original imprisonment might be called as witnesses, but Sirius could use his knowledge of their weaknesses to discredit them. And in the end, he is proven innocent, much to the joy of his supporters and the shock of his enemies.
Well, Sirius might face a series of challenges during the trial. He could be wrongly accused and have to fight hard to prove his innocence. Maybe his friends would come to his aid, bringing new evidence or testifying on his behalf.
One really good one I read had a detailed courtroom scene where Harry used some really clever magic to present evidence. It was on FanFiction.net. Another great one on Archive of Our Own had a unique take on how the wizarding world politics played into Sirius' trial.
Hermione would definitely be a great help. She's smart and resourceful, and could do a lot of research to find evidence. Also, Ron would be there, providing support and loyalty as always.
One possible plot could be that Harry discovers the truth about her real parentage early on. She then has to deal with the implications of being Sirius' daughter while still at Hogwarts, facing both the admiration and suspicion from others.
One possible reason could be a misunderstanding. Maybe some false information was spread about Harry that made Sirius believe he had done something unforgivable. For example, if there were rumors that Harry was involved with dark magic in some way, Sirius, who has a strong aversion to the dark side, might be quick to disown him.
One possible plot could be that James Sirius Potter has done something really naughty at Hogwarts, like pulling a major prank that went too far. The Howler then arrives, screaming at him in front of all his friends and classmates, causing him great embarrassment. This could lead to him having to face the consequences and also learning a valuable lesson about the limits of his mischief.
One possible plot twist could be that Sirius Black, being a wanted man, has to constantly move with Harry to avoid capture. This would make Harry's childhood very different from the one at the Dursleys. He'd grow up learning about the wizarding world on the run, perhaps meeting some unique characters in hiding places.
Well, perhaps Sirius trains Harry in advanced defensive magic. Harry could learn spells like the Patronus charm more effectively under Sirius's tutelage. Sirius might also teach Harry about the wizarding world's secrets that Dumbledore didn't share.
Obviously, Sirius himself is the central figure. His friends like Harry, Hermione, and Ron could be involved. They might be in the courtroom, trying to support Sirius in any way they can. The Ministry of Magic officials would also be there, as they are the ones conducting the trial.