As has been mentioned, there was no hereditary monarchy in Saxon England, though the most powerful families usually ensured their continued rule. For example, Alfred the Great followed his brother as King.
Edward the Confessor was not keen on Harold's family, the Godwin clan, because he considered them too powerful and it is possible he promised the throne to William in order to usurp their authority. Such a promise however would have had no legal standing under Saxon law.
Before Edward's death, Harold was involved in a ship wreck off the coast of Normandy/Brittany and ended up in William's court where the Normans made him swear an oath on a chest of Saints bones (relics) promising to support William's dubious claim to the English throne. Remember, William was illegitimate and any claim he had on anything, including Normandy, was always questioned and under threat.
After Edward's death, Harold was elected King by the Saxon lords who formed the Witan, who did not take any promise obtained under duress by William, seriously.
The invasion by another candidate for the throne, one, Harold Hardrada of Norway was supported by Harold's own brother, Tostig, and it was this invasion and the resulting Battle at Stamford Bridge, that seriously undermined Harold's ability to defend his kingdom, when William invaded in the south.