Nearly all the characters in Beowulf – in fact, we're willing to say that all of them – are concerned about establishing their own identities. Sometimes this literally means explaining who you are and where you came from to get other people to trust you. At other times, this means boasting about your own achievements and exploits in order to create a positive reputation for yourself. Before the evils of job-searching with a resume, medieval Scandinavian warriors had to "sell themselves" by talking themselves up, boasting, and making claims about their past victories.
Beowulf establishes his identity as a Geat warrior and eventually as a king through a combination of deeds, boasts, and gifts to his followers.
The most important task of a great warrior in Beowulf is to establish an identity so grand and legendary that his fame lives on after his death, in memorials and in ballads.