The Hunter is probably the oldest class in World of Warcraft. Before anyone in Azeroth took up an axe or sword, or learned anything of how to cast spells -- even before they learned to write -- they had to hunt for food. If they were like early Earth societies, the people of many nomadic groups would have relied on their hunters to bring in the meat they needed, as well as to protect the community from enemies. Back then, there would have been no such thing as fancy armor or complicated magical weapons. The relationship of a fighter to nature was just as important as the weapons he carried, if not more so.
Modern hunters in World of Warcraft come from the ancient tradition of those who learned to keep themselves and their families alive by living in harmony with nature. They learned the essential mysteries of survival in the wilderness, killing animals with stealth and primitive weapons, trapping them, and eventually turning predators and prey alike into friends and servants.
As time went by, those fighters who took up the path of the druid would learn to become nature itself; shamans would learn to call upon it; warriors and rogues would make battle their art, each in their own way. But hunters remained at that pivotal point between sentient races and the natural world -- they are connected to nature, but not manifestations of it; they work together with nature, but they do not worship it or call upon its spirits; they fight their enemies with the utmost passion, but they do it with the tools that hearken back to the dawn of civilization.
Beauty and the Beast
As a roleplayer with a hunter character, it's important to remember you don't play just one character -- you play two. A hunter may place more reliance on his own good aim, or on his survivalist wit, but in the end, everyone's got a pet to hold dear. Beastmasters place the most importance on this relationship with their pet, of course, but it's a significant part of any hunter's life.
Anyone who wants to roleplay a hunter would be wise to carefully consider the contribution the pet can make to the roleplaying experience, rather than just letting the creature follow him or her around all the time without anything to add to the atmosphere or story. Is this animal merely a more ferocious version of a house cat or family dog? What personality does your pet have? How might it respond to various sorts of situations? The idea here is not to create something so flashy that it steals the spotlight away from other players, but rather to give your pet just a bit of personality, so that it feels less like a machine on a leash, and more like a living, breathing creature that has actually become your friend.
You can do all sorts of things with a pet; the only limitation is your own imagination. A pet gorilla might love reading books. A pet ravager might play fetch and nuzzle people like the weirdest puppy you've ever seen. A pet raven could flap about perching on things, quothing "nevermore," and saying similarly odd things, like a parrot. A raptor might chew on any bit of furniture the hunter walks next to. A spider might wander about sniffing here and there, occasionally lifting its legs and leaving a little web behind to mark its territory. A core hound could get in a fight with itself (since it has two heads), accidentally try to run in two different directions at once, and generally behave as though one head doesn't know what the other wants to do (it can also drool lava on the carpets, which is fun). The possibilities are endless!
I highly recommend that every hunter come up with a story for how they tamed their current pets, as well as some small quirk or attitude that gives their pet the feeling of a minor supporting character instead of just a lifeless wallflower. You can use the "PetEmote" addon to make controlling your pets emotes easier (look here for some suggestions on how to use it).
Pets aren't the only supporting character in a hunter's life, though. His or her ranged weapon is almost as important, as it is the essential tool that defines the class just as much if not more than the animal pet. Like warriors and rogues, it makes a lot of sense for a hunter to take good care of his or her primary weapon.
But unlike warriors and rogues, a hunter's weapon isn't something you get into an enemy's face with. In fact, its usage requires hunters to stay as far away from their enemies as they can. Although some ranged weapons are big enough that enemies might be intimidated by them even at range, most hunters have bows, guns, or crossbows whose function is more important than fashion.
Consequently, a hunter's skill with his weapon has less of a showy feeling -- it requires patience, careful aim, and a steady hand to shoot well. The fact that fighting is an intellectual activity for hunters is reflected in the fact that hunters use mana, at least to some extent, and enhance many of their abilities with magical power. A hunter has to be a smart fighter, rather than just a strong or quick one.
I will survive
The civilized world is not necessarily friendly to hunters. Many innkeepers must be furious over the number of hunters who bring their pets inside and let them leave all manner of strange droppings all over the inns' fine carpets. How often would they shout at the hunter walking in the door with his hissing crocodile in tow, pointing them right back out the door again where the stable master is waiting to keep their pets safe and away from the scratchable furniture.
No, a hunter's true home is in the wild places of the world, and a hunter is uniquely suited to make the most use of his or her relationship with that world when confronting problems or enemies. Traps and Tracking abilities are two actual in-game skills a hunter possesses which mirror this relationship closely, and a hunter character should not hesitate to use them in his or her roleplaying. More than any other class, they allow you to turn the environment itself into an extension of your hunter's personality. Other survival skills, especially Feign Death, can give a hunter a lot of roleplaying flare when used in moderation, but be careful not to overdo it.
Something a little different
Of course, your hunter doesn't necessarily fit the stereotype. He could be an engineer who just loves his laboratory (and his guns) but doesn't really get out in the wild much; maybe his pet is just a random creature who snuck into his house one day looking for food, and somehow captured the affection of your character. Or, your character could be a trained markswoman in the Alliance or Horde military, much more concerned with rank, strategy, and protocol rather than animals and nature survival skills; maybe her pet is a rather sad and lonely creature, who visibly misses the old days when his mistress would pay more attention to him.
Daniel Whitcomb wrote an excellent article about the different archetypes you might choose from when moulding your hunter's personality and background, and all his ideas are excellent ones. You could adopt one of his suggestions, blend two of his ideas together, or borrow something from another class and work it in. Whatever kind of archetype you design for your hunter, the important thing is to try and give substance to what we see in the game -- let there be a reason for the pet, the crossbow, and the multi-colored traps your character has, and then chose whatever way seems best to you to make that reason shine.