Essentially his claim is that he was promised the throne by EdmundEdward the Confessor. But I was under the impression that succession was dependent on blood claim, i.e., descent from someone who held the throne or marriage to a woman - jure uxoris, or failing that, election (like how certain kings of france and holy roman emperors were elected). He was not elected, had no blood claim to the throne and neither did his wife. So why did people and nobility accept his claim?
When I was reading about English Kings and Norman dukes (on wikipedia but Idk where else to read for free) on Richard I of Normandy's page it says that his daughter's marriage to Aethelred the Unready and Cnut the Great added validity to William the Conqueror's claim. How? The two families did not wed again after that and he had no relation to english kings apart from that.
Secondly, he claims to have papal sanction but why did the pope help him?
Also what was the basis of Harald Hardrada's claim to the English throne?
EDIT: name
You are misinformed. William of Normandy was Edward the Confessor's first cousin once removed, and one of his closest living relatives. Edward's mother was Emma of Normandy, daughter of Duke Richard I, William's grandfather. In this regard, William had by far the strongest claim - neither Harold Godwinson nor Harald Hardrada were close blood kin to Edward the Confessor.
William's claim, as passed down to us by the Norman chroniclers, was built on three principles. As we've already established, he was a very close living relative to the deceased Edward. Second, he claimed that the childless Edward had promised him the throne several years before his death. Third, he asserted that Harold Godwinson had sworn (on holy relics, no less) to assist him in gaining the throne while he was shipwrecked/visiting in Normandy, and by going back on his oath to take the throne, Harold was violating the laws of man and God. This seems to have been the basis for the papal sanction. There is some doubt as to whether the last two justifications were strictly true. They may be Norman propaganda to retroactively strengthen William's claim; they may not be. We will probably never know.
Harold is supposed to have been named Edward's heir on the latter's deathbed, despite being very distant kin at best. What we do know is that the Wittan, a sort of council of the English nobles, proclaimed him king almost immediately after Edward's death. Harold was a powerful nobleman and a good soldier, and probably the most qualified English noble available for the job. This was the entire basis of Harold's claim.
Finally, I would caution you against relying too heavily on concepts of law and right as explanations for historical outcomes. There was no grand European court to arbitrate claims. Regardless of the veracity of his claim, William became king of England because he militarily defeated Harold and subjugated that part of the population which was inclined to dispute his rule; the average English peasant probably didn't care much one way or the other. It didn't matter if the English accepted his claim; it only mattered that he was able to convince enough of his Norman, Breton, and Flemish followers to support him.