<Stops to smell the flowers>
Although any class can be a lover of nature, druids are embodiments of nature itself. It only makes sense that they should appreciate the beauty around them, and even tend a garden or something to show how much nature means to them. If your druid is an herbalist, this can make even more sense. If you're feeling comedic, why not let your druid hug a tree now and then, just for kicks? or shift into animal form and talk to animals? or even take your friends to some tree-top or mountainside for a chat, rather than the plain old city tavern?
<Stops to smell the animal poop>
Second only to druids in their love for nature, hunters are the Warcraft class most likely to get their hands dirty. If you're a hunter, try to stalk something now and then, show off some of the freshest meats you've caught, or at least pat friendly animals on the head at every opportunity.
If you habitually walk around everywhere in the world with your pet hanging out, make sure to remark on the awkwardness of your wolf piddling on that pretty Silvermoon Inn carpet! Otherwise, you can either choose to put your pet away indoors, or else bring along a Poopie Purse.
<Studies a thick tome with a long and mysterious title>
To a certain degree, any of the caster classes are quite likely to be bookworms, but mages should top them all in obsessive bibliophilia. Mages should often mention something they read from Mysteries of the Arcane or A History of Magic or even Tubald Billywart's Exposition on the Three Uses of Implosive Energy Structures.
Likewise, a high-level mage should often take notes on his magical experiments. No matter how you use your magic today, you most likely learned it all through patient study. Why would you stop studying just because you're level 70 and know all the available spells? Have your character take an imaginative stab at which spells will be added to the mage class in the next expansion and start researching on them now!
"For the Light! Deny the Shadow! Defend the Innocent!"
Of all Warcraft classes, paladins seem most likely to utter an altruistic battle cry. Get RP Helper 2 and set it up with as many phrases as you can think of to be uttered now and then in combat (not so much that it gets annoying of course).
Likewise, when out of combat, go out of your way to do nice things for people. Pat someone on the back, say "Cheer up lad! Have faith in the Light!" and give them a blessing. Proactively offer some newbie some silver or gold and words of encouragement. Lay your hands on a child (who is standing still) and speak the words of a prayer for her well-being.
"A wise teacher once said..."
Priests in Warcraft may or may not be "religious" as such, but they should be repositories of cultural wisdom. Collect a good list of some wise sayings from your favorite spiritual tradition, edit them for general consumption and suitability within the Warcraft setting, and then share them with people whenever you get the opportunity. Don't hesitate to give pithy advice or wax philosophical. If you get any sort of audience, feel free to extend your sayings into short sermons on how life should be lived.
"Is that so?"
Rogues are supposed to be masters of hiding, so to a roleplayer, the art of playing a rogue presents us with an interesting opportunity: Many roleplayers (most people, perhaps) tends to be more interested in telling their own story rather than listening to yours. Therefore a great place to begin as a rogue is to become a great listener to the roleplay of others without making yourself the center of attention. Investigate their stories, quirks, hopes and desires, all the while keeping your own character in the background, until you feel as though you understand them well enough to land the perfect "roleplaying backstab!" -- that special word or action which can have a great effect on them and reveal some interesting taste of your own character's secrets too. As with any attack, don't be disappointed if you miss your RP backstab sometimes, when no one has the reaction you hoped for. Just re-stealth, listen some more, and prepare your next attack.
"Once, long long ago..."
Shamans are essentially priests of tribal societies. They should also give advice and wisdom to those around them, but perhaps do them in a different way. Since tribal wisdom was often carried on through oral traditions, shamans strike me as the Warcraft class most likely to be great storytellers and poets. You yourself don't have to be a great storyteller to make this work! Just collect other people's stories (especially fables), and adapt them for WoW.
If you're a player who is inclined towards playing the bad-guy, then warlocks are RP-easy mode. You can basically pick any villain from any old mythical story to use as inspiration, and then let your imagination go crazy with all sorts of nastiness.
Still, according to some roleplayers, warlocks aren't officially accepted in either Horde or Alliance culture, so you might benefit if you carry around some heavy books and pretend to be a mage. Be sure to grin evilly when people believe your lies.
<Polishes her sword>
Warriors can literally be anything or anyone. They have the vaguest stereotype and are the hardest to pin down to just one set of character traits. My warrior character is like a gnomish Indiana Jones, but yours could just as easily be an orcish General Patton. Whatever you choose, make sure that your warrior gets to show off some rage now and then, even if you keep it rather cold. Likewise, let your character show a special attention for his or her weapons and gear, since your class is most reliant on good gear, and you have to pay the most to repair it too.
What quirks do you use to make your character feel more authentic as a true representative of his chosen class?