Norman Conquest

William did not put forward any claimClaimants to the throne.
When the Confessor passed away in January 1066--right at that moment--there were three claimants to the throne:

1. Harold Godwinson, the Confessor's brother-in-law and the most powerful earl in the kingdom.
2. William, Duke of Normandy, the Confessor's distant cousin on the latter's mother's side of the family.
3. King Harold Hardrada of Norway, who claimed Cnut's heritage in England.

Immediately after Edward's death the Witan chose Harold Godwinson as king. To prove this point we can state that Harold was crowned  on the day immediately following the king's demise.

When Harold was crowned king in January 1066 he had to face the other two claimants, who decided to enforce their claims by force of arms and strive to become monarchs by right of conquest.

Harold Hardrada was supported by King Harold's brother Tostig, former earl of Northumbria. In 1065, Tostig had been driven into exile in Norway because of a rebellion in Northumbria. His elder brother had failed to support him and restore him to the earldom. In 1066 Tostig joined Hardrada in the hope of recovering Northumbria, taking revenge on his brother and--who knows?--seizing the crown for himself if he and Hardrada were victorious.

Hypothetically, then, after Harold was crowned king we might say that there were still three claimants to the throne:

1. William
2. Hardrada
3. (Tostig)

William's claim.

Before the Confessor died William did not put forward any claim, although he knew he had several:

1. He was Edward's distant cousin 

2. He stated that Edward had promised him the throne in the course of a meeting to which there were no witnesses.

3. When Harold had been shipwrecked off the Norman coast, been held prisoner by a Norman noble, and freed by William, the Anglo-Saxon earl had sworn an oath to the Norman duke promising to (we don't know): (1) become William's man-at-arms/vassal; (2) help William secure the English throne; (3) refuse to become king of England if chosen by the Witan.

After the Confessor died, when the Witan chose Harold (as it had the right and privilege to do because the nature of A-S monarchy was elective), William added a moral grievance to his previous claims: he would embark on a crusade to punish a perjurer. Whatever Harold had sworn, he had been bound to keep his word.

Eventually, William became king of England by right of conquest. Might proved him right.