This got asked a year or so ago and there are some pretty good answers in this thread:

https://www.reddit.com/r/AskHistorians/comments/2y0l4l/did_william_the_conqueror_know_that_harald/

:)

Most accounts of the battle of Hastings seem to suggest it was a close thing. Would William have had any chance if Harold's army wasn't drained from Stamford Bridge?

It's hard to say what Harold Hardrada knew for certain, but based on when he mustered his forces and when he invaded, it's fair to say that he was aware at the very least of the public activities of both the Godwinsons and Duke William of Normandy (who will later be crowned King William the Conqueror of England). In addition, after Tostig Godwinson's failed attack against Harold Godwinson, King of England, Tostig Godwinson fled to Harold Hardrada's protection. Harold Hardrada would have gained a great deal of information from Tostig Godwinson, and for what it's worth Harold Hardrada did bring Tostig Godwinson with him when he invaded England. However, Hardrada viewed Duke William of Normandy as an opponent, and therefore there was not any communication that we know of between the two, and it's unlikely that they kept secret counsel.

Duke William of Normandy kept a tight watch on the day-to-day events of the conflict. Not wanting to fight Harold Hardrada's forces nor wanting to be the first to fight Harold Godwinson, Duke William of Normandy mustered his army at the same time as Harold Hardrada, but did not cross the English Channel until Sept 27th, two days after Harold Hardrada's defeat at the Battle of Stanford Bridge. This allowed Duke William of Normandy to establish a foothold in southern England and forced Harold Godwinson to order his forces into a rushed march.

While Harold Godwinson did lose a great deal of his forces before encountering Duke William of Normandy's forces, it was not primarily due to the Battle of Stanford Bridge, although losses in that battle and previous battles had certainly whittled away some of his troops. Most of the losses before the Battle of Hastings were due to desertions between the Battle of Stanford Bridge and the Battle of Hastings. Harold Godwinson's forces were primarily drawn from the Fyrd, a peasant force loyal to the King of England. However, the march between the Battle of Stanford Bridge and the Battle of Hastings took place during mid-Autumn, the most important period of harvest, and crossed the majority of England. A great deal of his troops just decided to leave when they passed by their unharvested fields.

The Battle of Hastings was fairly interesting. Harold Godwinson was able to hold a hill for most of the day, in part through the longbow, a fairly new weapon with the ability to pierce through cavalry armor. That, combined with the rather swampy terrain, allowed the English troops to repel Norman strikes. However, Duke William of Normandy successfully feigned retreat, causing Harold Godwinson to break his shield wall in order to attack the retreating Norman forces. Duke William of Normandy ordered a charges against the now scattered English forces, winning him the battle.

I hope that answered your questions.

Source: worldcat.org/title/oxford-illustrated-history-of-medieval-england/oclc/36845492&referer=brief_results (The link should work, but worldcat assumes that I want my university's version of worldcat and I'm not super familiar with worldcat, so let me know if that doesn't work for you).