Would you consider Beowulf to be a proud or boastful person?

The short answer is: yes. Beowulf is indeed a proud and boastful man. But that's not because he's a conceited, arrogant man; it's more that he's living up to the expectations of his society. In the Nordic culture that Beowulf inhabits, it's not enough for warriors to perform noble and heroic deeds; they're expected to brag about them as well. They're expected to let everyone know just how incredibly brave, daring, cunning, and strong they are.
One such example occurs shortly after Beowulf first arrives at Heorot; no sooner has he been introduced to Hrothgar than he starts boasting about his "awesome strength." Later on, he goes one better and brags about how he'll face off against the fearsome Grendel without the need for weapons. He positively dares the monster to challenge him.
Even when Beowulf becomes king of the Geats he feels the need to assert his strength and skill as a warrior at every available opportunity. When a large, deadly dragon threatens his people, Beowulf doesn't hesitate to strap on his armor and get ready to do battle. But this isn't because he wants to stop the dragon from killing people; it's because he wants to win, to prove himself once more on the field of battle. For Beowulf, as with all Nordic warriors, it's all about achieving glory and boasting loudly about it afterwards.