In addition to the excellent answer above, the Anglo-Saxons practiced pagan revenge and believed in pagan concepts of wergild (man-price) and the wyrd (fate, destiny). Also, the poem mentions that there is no afterlife.
These pagan concepts go against Christian doctrine ("love your enemies" and "turn the other cheek"), but the monks had to leave them in because they are so prevalent throughout the poem. These pagan concepts relate to the brevity of human life, the inability to control one's destiny, and the cyclical nature of revenge and violence.
Example of Wyrd are in lines:
To Hygelac send, if Hild should take me,
best of war-weeds, warding my breast,
armor excellent, heirloom of Hrethel
and work of Wayland. Fares Wyrd as she must."
AND for Wergild:
Heorogar was dead,
my elder brother, had breathed his last,
Healfdene's bairn: he was better than I!
Straightway the feud with fee I settled,
to the Wylfings sent, o'er watery ridges,
treasures olden: oaths he swore me.
(this "fee" is the money paid to settle the fued. It is the wergild, or man-price)
THE fall of his lord he was fain to requite
in after days; and to Eadgils he proved
friend to the friendless, and forces sent
over the sea to the son of Ohtere,
weapons and warriors: well repaid he
those care-paths cold when the king he slew.
For No Afterlife:
...all of us with souls, earth-dwellers/and children of men, must make our way/to a destination already ordained/where the body, after the banqueting/sleeps on its death bed
Beowulf was transcribed by Christian monks. The Anglo Saxon legends were passed down through oral tradition, not written down, so by the time the monks got around to transcribing these legends, they interspersed their own Christian values into the pagan poem. This is why the poem seems to combine these two opposing elements.
First of all, the culture of Beowulf values heroes, not a single deity, as in Christianity, so this is a pagan practice. Further, making vows of sacrifices at idol fanes, paying attention to omens and burning the dead are all practices that were frowned upon by the monks. The Anglo Saxon practice of decking out a corpse in all its finery, with rich "earthly treasures" is also contrary to Christianity and very similar to what the Egyptians did with their mummies.
Beowulf, on the other hand, acknowledges God throughout the poem. He thanks God for guiding and protecting him, and states that in his battle with Grendel, he would have been destroyed if God had not guided him. Elsewhere in the poem, he also acknowledges that all earthly blessings come from God. Other characters in the poem also acknowledge that power comes from God - Hrothgar in particular when he talks with Beowulf about selfish kings who do not give thanks to God for their blessings.
Read about the poem right here on eNotes. You can also read the text online.
I agree with the first editor that Beowulf is a poem that reagrds the Christian faith as being the primary faith. However, when King Hrothgar continues to send out warriors who get killed, and Grendel continues to kill off the people in the kingdom, he and his clan build alters and try worshipping pagan style in hopes that it will help as well. He is desperate to try every measure to find relief from GGrendel.
"Whiles they vowed in their heathen fanes
altar-offerings, asked with words [footnote 5]
that the slayer-of-souls would succor give them
for the pain of their people. Their practice this,
their heathen hope; 'twas Hell they thought of
in mood of their mind."
Hrothgar's men were aware that what they were doing was against their own faith in God, but they were desperate to find peace from Grendel. Bribery had not worked either.