In the epic poem, Beowulf, I have not found reference to Beowulf's darkness as a character, but references to Grendel abound. (References to darkness regarding Beowulf the character are associated with the time in which he lived.)
Grendel is a creature who is alienated from the comforting company of men, their fires, their songs, and their lives. He is believed to be a descendant of Cain, Adam and Eve's son who murdered his brother Abel. For this sin, Cain was marked and driven from the land. The poem states that Cain spawned a line of monsters, of which Grendel is one. Besides the fact that he is cursed, Grendel is lonely and resents the camaraderie he witnesses at the mead hall. When his fury becomes too great, he attacks the mead hall, and continues to do so for twelve winters. This destruction is the only thing that brings the monster any sort of perverse pleasure.
In terms of the darkness of Beowulf, we might find a darkness in the fact that Beowulf's fame is derived from fighting wars and men (though in the story, only the deaths of two men are mentioned). Bloodshed is a way of life for the men of this era. Age comes upon one too quickly. There may be a sense of darkness based upon how short life is (how quickly it passes)—
Youth leads too quickly to decline...
...and the brutality life brings, as mentioned above. Being a hero may be a full-time job, but it is not a life-long career: it is only open to the youngest and strongest of warriors. The poem itself has a sad tone:
...commentators have perceived in the somber, elegiac tone at the close of Beowulf the culmination of such themes as aging, the destructive and endless nature of feuding, the shortness and brutality of life, and the death of the pagan heroic code—a system of belief which offered immortality only through fame.
The third part of the poem may find Beowulf to be sadder as fifty years have passed and he is called upon again to fight, this time to protect his subjects, as King of the Geats. The sadness is not in the fight or even in the danger, but perhaps most of all in that the men who serve him later in life have lost a sense of valor and dedication, which was a very real and present element in Beowulf's younger life when his men followed him unquestioningly into battle. In fighting the dragon, however, all but one of Beowulf's men flee: Wiglaf—who demonstrates the same characteristics Beowulf did as a young warrior.
In any story, but especially in such an old tale, good vs evil is present, seen between hero and monster, or the presence of light and darkness. I believe the darkness we perceive in this story is found in the later part of Beowulf's life as he sees the changes life has brought, not just to him, but to the sense of honor and bravery exhibited (or not) by many of the men who serve him.