It might be interesting to make Beowulf's job something that is unrelated to war and survival. After all, many of his traits - independence, tenacity, strong sense of duty, compassion, leadership - are traits that would be helpful in any number of jobs.
For example, wouldn't Beowulf make an interesting teacher? Or perhaps a company president? Or even a salesman - consider his determination, especially when holding on to Grendel's arm.!
Once you've picked your job:
1. Start with a salutation (Dear ________,)
2. Have Beowulf introduce himself and his main qualities.
3. He should then announce what he is writing for.
4. Next paragraph, go through why he is good for this position. List examples from his "previous jobs"
5. Final paragraph, explain why he is interested in the job and thank the reader.