This ain't your daddy's death knight
But before we go on, you should be aware that there are two kinds of Death Knight in Warcraft lore. The first kind was actually more of an undead necromancer -- a powerful orc warlock whose soul had been placed inside the fallen corpse of an Azerothian warrior. Some looked mostly normal except for being a little pale, while others were little more than skeletons; but all of them had tremendous magic and retained their free will. The most famous of these was Teron Gorefiend.
That is not what you are though. You are part of a new breed of death knight based on Arthas' unquestionably successful prototype. You were once a normal hero of your people, but somehow were forced into the service of the Lich King, most likely by dying in some sort of battle with the Scourge. If the Lich King and his minions believed you to be worth the trouble, they reanimated your body and overruled your own free will. You eventually get your free will back, of course, but the undeniable fact remains that for a time, you were a servant of the Lich King, and you did his horrific bidding.
Some death knights joined up with the Lich King out of their own free will, in spite of the fact that this meant they wouldn't have free will anymore. Arthas, of course, was so hell-bent on getting vengeance that he eventually went insane and didn't care who he killed as long as he could inflict as much death and destruction as possible. Baron Rivendare is another example of someone who thought being a death knight would be pretty cool. Your character is much less likely to have actually chosen to become a death knight like this, simply because, if you wanted to serve the Lich King so much, you wouldn't leave, even if you had the chance? Our death knight characters may be rotten scoundrels, wicked to the core, but for whatever reason, they don't like the Scourge very much. They were probably actually good people once, and now that they have their free will back, for the most part, the Knights of the Ebon Blade (of which your character is one) seem to want to go back to being good insofar as it is possible to do that when you're an undead master of necromantic rune magic.
So what would it have been like to be under the spell of the Lich King like that? What memories of those dark days would your character carry around? The easiest way to deal with these issues is to say that your character has no memory of that time -- you were possessed and therefore not at all responsible (even to yourself) for any of your actions while under the influence of undead mind control. This may be the best possibility for some characters, especially those who want to maintain more innocence and goodness, but to some roleplayers it will feel like a cop out. Should we go deeper?
From my experience playing through the death knight starting zone, it didn't seem to me like my character (or any others, for that matter) was under the complete control of the Lich King. That's to say, of course the Lich King told me what to do and I had to obey, but I always got the feeling that the exact way in which I obeyed was really up to me. I felt as though I almost had enough strength to actually disobey and just let myself be cast aside as a sort of martyr, possibly like one of the other "unworthy initiates" you see in that starting area. There are also small choices you can make which the Lich King doesn't seem to be entirely aware of (such as your choice to listen to your old friend rather than just dispatch him or her right away).
This sense of limited free will makes me feel as though my character should remember every bit of those dark days, every evil action committed, as if it were his own. Naturally, in his mind he understands he could not fully control himself, but in his heart he would feel guilty for all that senseless killing he wrought with his own hands. Now, he would want to redeem himself. Even though he can no longer go back to the way things were, take up his old magic and fight as one of the living, at least he can use his new powers, his curse, for the greater good.
That is far from the only option, however. Another character could simply wander about the world rather lost, actually missing the voice of the Lich King in his or her mind; perhaps the original identity has been mostly destroyed and all that remains is this shell that does pretty much whatever other people tell it to do. Someone else could be a borderline psychopath, such as a former priest who used to clean up after everyone else, healing wounds and such, and now takes a bit too much pleasure in his or her own killing sprees -- all the while convincing him or herself that it's okay to enjoy killing this much because he or she is on the side of the good guys now. Yet another character may feel that his or her actions under Scourge command were perfectly acceptable because those Scarlet Crusaders were evil and deserved to die anyways; sorry about the mothers and peasants that died, of course, but honestly they should have been over in Stormwind with the good humans rather than hanging out near Tyr's Hand with a fanatical terrorist organization like the Scarlet Crusade. The truth of these things is not so simple, of course, but in the end what really matters isn't what's true, it's what your character thinks is true.
A small note about peer pressure
What other people think is true about you is another story altogether though. Naturally the opinions of society will have a great impact on your character's own impression of him or herself, too. Thrall and Wrynn have each commanded their people to show you respect, but likely that respect doesn't go very deep for most common folk. Do they say "hello sir" and sell you their wares, all the while sneering at you behind your back? Does everyone you meet cower slightly at the sight of you, or treat you like just another shmuck with a sword walking around town with a bad attitude? All that is up to other roleplayers on your server of course, but your reaction to it is up to you.
All the World's a Stage continues this series on roleplaying within the lore with today's look at Death Knights. Be sure to check out Priests, Mages, Warlocks, Paladins, and Warriors (Horde and Alliance). For more about roleplaying the different classes, see how spells themselves can be used in roleplaying, for Druids, Hunters, and Mages, as well as Paladins, Priests, Shamans, Warlocks and Warriors. Also, read why all races can be death knights.