Although I don't play on an RP server, whenever I create a new character I feel compelled to come up with a backstory that explains their purpose in the world. One of the factors that drew me to playing a Death Knight is that unlike every other hero in Azeroth, the one thing we know for certain about this character is that he died. He fought the Scourge and lost. This adds layers of grief and vulnerability that previously only Forsaken characters could relate to.
For Terathis, I picture a scene where he rushes home from the front upon hearing that his home village is being attacked. He is able to find his wife and children, but his relief at finding them safe is short-lived. A group of Scourge breaks down the door of their home and he orders his wife to take the children out the back way. He fights bravely, but meets his end on the sickle of an Abomination.
So now his personal quest is two-fold. He seeks vengeance against the Lich King, but also to discover what happened to his family. He has to live with the tragic results of his own failings and try to salvage what he can of his past life.
None of this has much of an effect in game, although I have decided that he will never pursue any professions other than First Aid because it would take time away from his search. That's about as far down the RP rabbit hole as I'm willing to go. I don't speak in character, or anything like that. But I like the feeling that this avatar, this person if you will, that I'm going to spend hundreds of hours with has a story worthy of that commitment.