There’s a lot of debate on the subject! I personally have a “do whatever the heck you want, all character circumstances are different” approach to the issue, as long as you take into account what the character has gone / is going through.
They were expressly conditioned to be monstrous killing machines void of any remorse for their actions, but that doesn’t mean they were stripped of unique personalities. What we see of them personally in the Scourge ranges from cold and emotionless, to snarky, to guilty regardless of that training, to gleefully enjoying their jobs, to just kind of all-around angry and cruel. That’s already a pretty varied set of emotions even under that banner, so for those without the Lich King constantly whispering murder in their ear, they might have a greater chance to explore those in-depth.
There’s an entire subplot dedicated to Thassarian and Koltira, characters for whom a mutual respect became a genuine friendship. They’re probably the most personally explored death knights outside of Arthas himself, and are good case studies for how with a strong enough will, the Scourge isn’t always capable of completely emotionally stripping its victims. Thassarian was willing, while in the Scourge, to put his duties on hold and carve a path through some Scarlet Crusaders just to recover Koltira. After their freedom, he learns that Koltira is being haunted by shadows, and expresses a desire to help him with no other motive than that’s what friends do. Their friendship caused problems when they had to face each other on the battlefield.
Here’s Thassarian straight-up crying after realizing what he’d been doing in the Scourge:
(sorry, had to use my phone for that— scanner’s broken)
When he sees the Scourge has captured his sister, he loses his shit. When he rejoins the Alliance, literally his first action is to go sulk in a bar and drink.
Talking to either character in-game, the overall vibe they give is “angry”, but considering what they’ve been through and what they continue to experience, it’s not particularly unreasonable. They’re definitely softer than some other death knight NPCs, and they’ll make an occasional morbid joke.
I’ve kind of strayed a bit but I guess the point I’m trying to get to is: it entirely depends on the character.
Ask yourself: who were they before they died? Did they willingly join the Scourge, or were they forced into it? How good are they at coping with what they are now? How interested are they in pursuing what they were before (there is, apparently, a DK follower in WoD that just happily loves to fish)? These are all things to take into individual consideration.
As a whole group, they’re mostly just kind of grumpy and bitter, with some enjoying their new selves a lot more than others. Individually, I don’t think it’s fair to say they HAVE to be any one thing, because there are plenty of examples saying that’s not true! You’re probably not going to see many death knights that spend their not-killing time baking cookies and singing camp songs with orphans, but to say they’re incapable of experiencing any flavor of joy or love is, I think, incorrect.
Hope this helps. C: