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...

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'

Great
king.

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:

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...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.