The promise and the bloodline were the basis, it gave him the pretense of entitlement with which to go after the throne with a measure of legitimacy. However, he claimed the crown more through right of conquest than on the basis of any bloodline.

What eventually happened with the throne makes more sense with context. Edward the Confessor was the son of Aethelred and Emma, Emma is the great-aunt of William. When Cnut took the throne, he married Emma and sired children by her. One of the men of Aethelred’s court who hadn’t taken Cnut’s side was Godwin of Wessex. He managed to win Cnut’s favor and ultimately was created Earl of Wessex. When Cnut died and a power struggle ensued, Edward the Confessor made peace with his half-brother and took the throne when the brother died. Edward married Edith, daughter of Godwin, but he never sired children by her possibly out of spite to the ambitious Earl Godwin. The Earl grew more powerful but at one point Edward managed to banish him from court for a time. This was the occasion it was written in the Anglo-Saxon chronicles that William came to England. William and Edward spoke privately, and according to William, Edward promised him the throne. Soon after Earl Godwin was taken back to court and grew increasingly more powerful, enough so to threaten the king. The Earl Godwin died, then Edward died without legitimate children. Harold Godwinson claimed Edward had promised him the crown, William claimed the same, and Edgar the Aetheling who was nephew to Edward was considered the natural heir. However, Edgar was seen as too young, while William was in Normandy and his claim wasn’t seen as particularly great. Harold Godwinson didn’t have the royal blood, but he had plenty of power as Earl in his own right. The Witan, a Saxon council, had the right to declare the heir and saw the safest option as Harold.

Eventually William got hold of Harold and made him swear on holy relics to recognize William as king and further his claim. Harold reluctantly swore the oath which he later broke. William had also gotten himself a bride against the wishes of the Pope, one Matilda of Flanders. A former lawyer turned monk Lanfranc opposed the marriage initially but then came around and convinced Pope Alexander II to grant dispensation for the marriage. Lanfranc assisted William in making a case to the pope for why he should be king. Prior to this, the Godwin family with their hands in every cookie jar had been favored by the church. William pressed his claim and also emphasized that Harold was an oath breaker on a vow sworn with holy relics. William got Harold excommunicated, which at the very least meant the church wouldn’t support him. That didn’t mean necessarily that they would support William either. But with Lanfranc’s assistance William reached an agreement with the Pope. The promises William said he made and the promises the Pope said William made differ, but it was understood that at the very least William was supposed to be a benefactor to the church.

At the end of the day, no one had a strong enough claim to pull the divine blood right card to great effect. Edgar came the closest but he was too young to be reasonably considered. William with his snippet of blood claim had a huge army and was prepared to continue fighting for the throne regardless of the Pope. Harold was an oath-broker and had committed a heresy, and had no blood claim. William made assurances to the Pope, whereas Harold had not. Ultimately it wasn’t a matter of the Pope deciding William’s claim was virtuous, it was a practical decision that was influenced by Lanfranc’s mediation. For William’s part, the blessing of the Pope essentially made his conquest a holy war, validated his claim, and integrated the Normans into everyday fabric of the people of England.