The character of Beowulf is probably the earliest example of what we would consider a hero. There isn't much difference between him and somebody like Captain America. In Beowulf's situation, like in modern hero movies, there is a great evil that exists, and all existing law enforcement is incapable of dealing with the evil. Additionally, diplomacy either has failed or simply won't work at all. Finally, the evil that is present is somehow pure evil and seemingly indestructible. We see that with Grendel in that he murderously rampages through the hall night after night, leaving a bloodbath in his wake. As for his "evilness," he is a descendant of Cain, who killed his brother in the Bible. No existing warriors can handle Grendel, so a special, equally strong and indestructible hero is needed to violently vanquish Grendel and Grendel's mother and wipe them from the face of the planet. We know that Beowulf is quite strong, based on his appearance alone.

Nor have I seen
A mightier man-at-arms on this earth
Than the one standing here: unless I am mistaken,
He is truly noble. This is no mere
Hanger-on in a hero’s armour.

Beowulf fights Grendel and Grendel's mother because he's a strong, brave warrior, and nobody else is able to do it. Fighting evil like Grendel is his duty, just like fighting Thanos is the duty of the Avengers. It's the same setup. Beowulf is expected to do this for the simple reason that he is a hero, and heroes selflessly put themselves on the line for the good of others. Make no mistake—Beowulf is a selfless hero. He is offered the Danish throne and riches beyond his imagination, yet he turns it down. Defeating evil and earning the respect of his fellow men is payment enough. Regarding the question as to whether or not he accomplishes his goal, I believe that the answer has to be "yes, he does." The goal is to eliminate Grendel and Grendel's mother, and that is what Beowulf does.

Although Beowulf's first two fights—the fight against Grendel and the subsequent fight against Grendel's mother—accomplish similar goals (i.e., ridding the Danes of a monster), he enters into them for different reasons. Beowulf's reason for fighting Grendel heavily involves societal expectations, whereas his fight against Grendel's mother appears to stem primarily from a sense of responsibility to protect Hrothgar and his people.

Beowulf undertakes his journey to confront Grendel for two main reasons: to win glory and to pay Hrothgar back for aid the old king provided to Beowulf's father. Both of these were things expected of warriors in Anglo-Saxon society, and Beowulf fulfills these expectations by undertaking his fight against Grendel and defeating the monster.

The problem presented by the attack of Grendel's mother after Grendel is killed is a different issue. Here, Beowulf has already accomplished what he initially set out to do, so he could leave it to the Danes to fight Grendel's mother on their own. However, Beowulf seems to see this second fight as a duty he personally owes to Hrothgar, as the king is relying on him to save the Danes. In killing Grendel's mother, Beowulf is honoring the newly created bond he shares with Hrothgar. In other words, he is acting as a thane, a retainer of the king, even though he is not truly Hrothgar's retainer. This stands in contrast to when he first came to Heorot as a warrior from the outside who was seeking renown and to pay a debt.

By undertaking the fight against Grendel, Beowulf accomplishes his objectives to win renown and to pay a debt owed to Hrothgar. However, when he fights Grendel's mother, he is doing so because he feels responsible to ensure the safety of Hrothgar and his family and people. His objective is to finish the job and rid the Danes of Grendel's mother, and he does so by slaying her.