Walk through the door
Different roleplayers can give you different answers, but many revolve around the same principle: A regular player plays the game to overcome challenges, get rewards, and chat with people; whereas a roleplayer plays the game to relish in creativity as well, to perform along with other performers. In short, a roleplayer is seeking an artistic experience in addition to all those other things that a normal player wants. The game is not just a game, but a canvas.
When you log into the game, put the world and all its concerns aside for a while, and log into your character's personality as well. Hold the idea of who your character is firmly in your mind and don't let go. Live it and breathe it for a while as your very own artistic creation, interact with others in the game from this new and creative state of mind, and then when you've had your fill, start making your way towards logging off, or doing something else out of character for a while if you want. You don't have to be in-character all the time unless you choose to be (though it is important to respect the roleplaying atmosphere others around you may wish to maintain).
I write this because I have sometimes found myself and others around me just typing out the same type of common gaming communications we often use in dungeons or elsewhere, only saying it in a different accent, or without reference to gaming lingo. The words we use are technically "in-character," but actually there's no real character depth in them. At these times, roleplaying loses its artistic element and becomes a mere form, bearing the semblance of art but not the substance of it.
Find out how deep the rabbit hole goes
To counter this problem, it is helpful to create a solid framework in your mind as to who your character is -- something much more than just a vague impression or idea. While such a general character concept is a great place to begin, after some time you must settle down on a permanent design and add interesting details. It's a lot like the process you go through when you take up a piece of clay and begin molding it to the shape you have in mind. Before long, you find a solid shape is emerging, and it's better than what you thought it would be.
Once your character is very clear and solid in your mind, it is much easier to log into and out of his or her mindset and worldview. You will find new and interesting motivations and solutions to problems arising naturally in your interactions with others and with the actual game activities as well.
There are many ways to further solidify your character and understand his or her mindset more deeply, each of which deserves its own article. One approach is to create a kind of map of your character's personality, either by writing down all your ideas in a notebook, or by using a more organized method such as a "Character Diamond."
Another approach is to sit down with an encyclopedia of Warcraft lore and try to map out a life story for your character. This story isn't so much something that you will tell to anyone who will listen -- rather, it is a tool to help you know how your character feels and acts in any given moment. It can and should be vague in some places, so that you have plenty of freedom to add to it in the future as opportunities arise, but detailed in others, so that you can draw on richly imagined experiences while thinking of your character in the present. You may find lists of character questions and other such exercises very useful in focusing in on the important details.
Whatever method works for you, remember not to let your precious character drift away as life gets busy and the pressure to have the latest epic gear increases. Invest a bit of effort in putting real life aside for a short time, and get inside your character's mind. Play your role in such a way that others are drawn in with you, into your imaginary world.
Edit: I modified some links that apparently required forum access at Dramatis-Personae. Fortunately they have some of the same information posted in general character pages too.