Quote 1
"I had a fixed purpose when I put to sea.
As I sat in the boat with my band of men,
I meant to perform to the uttermost
what your people wanted or perish in the attempt,
in the fiend's clutches. And I shall fulfill that purpose,
prove myself with a proud deed
or meet my death here in the mead-hall." (632-638)
It's all or nothing in this fight to the death: the good warrior Beowulf against the evil demon Grendel. Things can't get much more clear cut than that.
Quote 2
"I have wrested the hilt
from the enemies' hand, avenged the evil
done to the Danes; it is what was due." (1668-1670)
The battle between good and evil is a necessary part of Beowulf's life; it consists of fighting for justice, for "what was due" to a people who have suffered wrongs. Notice that, in this passage, good is not just the opposite of evil – good is actually the process of avenging evil that has been done in the past. That's a dangerous belief, because it leads to unending feuds and wars among the different Scandinavian and Germanic tribes.
Quote 3
"I had a fixed purpose when I put to sea.
As I sat in the boat with my band of men,
I meant to perform to the uttermost
what your people wanted or perish in the attempt,
in the fiend's clutches. And I shall fulfil that purpose,
prove myself with a proud deed
or meet my death here in the mead-hall." (632-638)
Having explained who he is by explaining what he's done in the past, Beowulf stakes his identity on what he's going to do in the future.