What do we learn about Beowulf's character from his speech in lines 365-405 in Beowulf?
The speech begins:
Hail to you, Hrothgar! I am Hygelac's
kinsman and comrade, esteemed by the king
for deeds i have done in the years of youth.
I heard in my homeland how grendel grieves you.
Seafareers says that your splendid hall
stands idle and useless after the sun
sinks each evening from Heaven's height......
OK, thank you so much again!
What you are asking about is characterization. What characterization is accomplished by Beowulf's speech?
First, we learn that Beowulf has a sense of duty:
My people have said, the wisest, most knowing
And best of them, that my duty was to go to the Danes'
The mead hall is plagued by Grendel, and Beowulf feels it is his duty to come help.
We also learn that Beowulf has performed great deeds: rising out of the darkness of war covered in his enemies' blood, chaining five giants, etc. His deeds are his testament, his demonstration of worthiness.
Beowulf also believes in fate, at least when he is using rhetoric in the form of a speech:
...Now Grendel and I are called
Together, and I've come.
His battle with Grendel is his destiny, his fate.
Beowulf is also independent--he requests to fight Grendel by himself--and honorable--he will use no weapon, because Grendel uses no weapon.
Finally, Beowulf puts himself into God's hands. He leaves it up to God to determine the outcome of the battle.
Scholars assume that these personality traits and beliefs were important to Anglo-Saxons, in general. Beowulf is probably the ideal Anglo-Saxon hero.
One piece of characterization that encompasses some of the elements in the previous post is Beowulf's enormous ego and bragging speach. Modern audiences are sometimes put off by this bahvior because we tend to value a humble attitude, but in the Anglo-Saxon culture, reputation for bravery and skill in battle were vital to life, so Beowulf's listing of his past accomplishments and his determination to fight alone and without a weapon are part of this culture. He doesn't play coy; he directly states that he has accomplished many great deeds and goes on to list them. This kind of talk was meant to both intimidate foes and reassure the people -- they learn first hand what Beowulf thinks he can accomplish, and stay tuned to see if he can deliver what he promises.