How does Beowulf best exemplify a hero in the work, Beowulf?
There are several ways in which Beowulf exemplifies the image of a hero in the work. Let me name three.
First, Beowulf comes from a distant land. According to the poem he comes from Geats and he helps the beleaguered king of the Danes. This distance is an important part of what it means to be a hero. Keep in mind that the heroes of the Greek and Roman worlds come from distant places a well. Achilles is the king of the Myrmidons, Odysseus is the king of Ithaca, and Aeneas travels from Troy to establish Rome.
Second, Beowulf shows courage in the face of danger. He does not even care for his own life. Keep in mind that he kills Grendel with his bare hands, because Grendel is immune to human weapons and he kills Grendel's mother in her lair alone. The individual duel is also a mark of what is means to be a hero in some ways. This can be seen in the duel between Hector and Achilles in the Iliad and Aeneas and Turnus in the Aeneid.
Third, Beowulf dies in battle in this fight against the dragon. This is another mark of what it means to be a hero in Western literature. The greatest hero of all, Achilles, also died in battle. In fact, according to the story, Achilles went to Troy knowing that he would win great fame, but an early death.