Beowulf is certainly brave and strong, but I would argue that his perseverance is his strongest characteristic. He keeps going even when it is difficult, and that is ultimately why he succeeds.
Beowulf has a great sense of pride and will not leave a battle unfinished. This is revealed in big and small ways throughout the text. He kills Grendel, but he makes sure that the battle is done by taking Grendel's arm and nailing up on the mead hall front door. He then doesn't shy away from Grendel's mother who could be just as menacing -- he finishes the battle of Grendel by eliminating any threats from her as well, putting the final touch on this conflict.
One thing we see about Beowulf is his desire for fame- or to be remembered. When de dives into the lake to fight Grendel's mom, the book even tells us, "Beowulf, longing only for fame, leaps into" the lake. Even our modern day heroes may pause and think about the best plan to fight the monster's angry mother, but not Beowulf. Pausing would make it seem as though he doubted his own abilities, and so he grabs Hrunting and a helmet, throws on his mail shirt and dives in ready to fight.
One of Beowulf's most important characteristics is his desire to help others, even at his own expense; this is best seen in the final battle with the dragon, where despite his advanced age he lays his own life down to defend and protect his people.
Beowulf displays a number of important characteristics, including bravery, intelligence, initiative, concern for other people, loyalty to his superiors, loyalty to God, humility, and, of course, great physical strength as well as real skill in the use of that strength. Practically all of these traits can be seen in the description of his fight with Grendel.