At the end of Harry's third year, Harry, dissapointed that he couldn't do anything to prove Sirius was innocent, decides to confront Dumbledore and tell him he doesn't want to go home to the Dursleys. He tells Dumbledore about how they treat him badly and how he was looking forward to living with Sirius only to not being able to show Minister Fudge Sirius was innocent.

Dumbledore, dissapointed that Petunia didn't do what he asked, decides to tell Harry about the prophecy and why he was placed with the Dursleys. He then convinces Harry to allow him to use a spell that will send both of their souls back to shortly after Harry became an orphan.

Dumbledore, prevents Sirius from going after Pettigrew and manages to keep him out of Azkaban. He allows Sirius to raise Harry (who, unknown to anyone but Dumbledore is mentally almost 14 years old) under a few conditions:

Sirius passes Harry off as his own son and tell every one that Harry died. Sirius can tell Remus the truth but that is it. Dumbledore will help him by making Harry biologically Sirius' son.

Dumbledore, not wanting anyone to be suspecious of Harry being from the future, uses the soul fragment Voldemort accidentaly put in Harry to make Harry a twin brother.  He does this so Harry can get an idea of how Harry's physical age should act. Dumbledore must use his knowledge of magic to make this "child" he created not be a horcrux but a seperate person who Voldemort is not depended on for immortality and Sirius has to agree to pass this child of as his own as well.

Things go smothly until the child that Dumbledore created starts having dreams about being Voldemort. Sirius, Remus, Dumbledore, and Harry work together to try to prevent the child from fallowing in Voldemort's footsteps while trying to defeat the real Voldemort.

Must Haves:

Sirius gives Harry and the child Dumbledore created Black family like names. (The Blacks have a tradition of naming their kids after characters in mythology)

Sirius and Remus must not know that Dumbledore and Harry came backwards in time until after Harry starts Hogwarts for the second time.

The world will be forever convinced that Harry Potter died with his parents (Voldemort can somehow find out the truth).

Dumbledore tracks down Voldemorts horcruxes.

Harry and the child Dumbledore created are both parslemouths, even after Voldemort dies.

Due to the apparent deaths of all the Potters, Sirius inhertits everything.

Optional Things:

Dumbledore takes steps to prevent the events of Harry's first two years of Hogwarts (IE, not allowing Quirrel to teach again, having students pocessions being checked for dark magic).

Dumbledore convinces Snape to be nice to students (espessaly to Sirius' supposed kids).