He was a member of the Bellman's Crew, which was of course led by Bellman. This expedition hunted Snarks. However, when his comrade, the Baker, apparently found a Snark (which was a Boojum), he disappeared. The expedition was ended after that.
The Beaver was created by Lewis Carroll in his 1874 nonsense-poem "The Hunting of the Snark". He was illustrated by Henry Holiday.
In 2002, Alan Moore took up the crew in the second volume of League of Extraordinary Gentlemen. Instead of hunting for Snarks, they were an expedition who ventured down the rabbit hole. In this version, the Beaver is actully a woman named "Miss Beaver". Miss Beaver was among the members of the crew which returned insane. She is only mentioned. Only the Bellman appears.
In 2011, Roger Langridge created a comic called Snarked, which was based on Carroll's original poem. In the #0 issue, Henry Holiday's original illustrations were reprinted along with Carrol's poem and the Beaver appeared in a few of those illustrations. In issue #1, Langridge retold Hunting of the Snark in his own words. However, it was not until issue #3 that the Beaver and the rest of the crew joined the cast of Snarked.
Major Story Arcs
Bellman Expedition (League of Extraordinary Gentlemen)
Miss Beaver was a part of the Bellman Expedition, which went down the "Rabbit Hole" like Miss A.L. She and the rest of the crew did return, but they were insane.
Following the events at Snark Island, the crew returned home. They were considered the laughing stock of the town because no one believed their story. Even the Baker's brother, the Barber, had his doubts. In order to get any work, they changed the name of their ship from "Snark Hunter" to "Old Gratitude".
When Wilburforce J. Walrus and Clyde McDunk needed a ship to take Queen Scarlet and Prince Rusty in search of her father, the Red King. Only Old Gratitude and her crew would take then. Of course, this was before they knew where they were going. However, they were persuaded by Mr. Walrus, who is a conman, that their friend the Baker had not disappeared forever but merely was sent forward in time. Believing his tall tale, the crew decided to continue taking the Walrus and the Carpenter and royals to Snark Island.
On their journey, they have faced the Mad Hatter's pirate crew and the fiendish Gryphon. Most recently, they have reached Snark Island and have saved the Red King, but they are now stranded on the island.