There are a few things that make the DK stand out from other classes in the game.
Firstly, from a class fantasy standpoint, they fill the role of the dark warrior/overlord archetype, utilizing powerful armor, weaponry, and dark magics in battle. This archetype is used by several very cool characters, not the least of which is Arthas himself but also pertaining to characters like Darth Vader and Sauron. Indomitable, inexorable forces of darkness that slowly advance and crush anything in their path. Anyone who wants to play a character like this will often be drawn to the death knight.
Looking a bit closer, we can see what more specific abilities the Death Knight has. Not only do they possess the abilities of a necromancer, but also mimic the abilities of plenty of other badass undead creatures, such as vampires, undead dragons, and liches. All combined into a single powerful warrior, who, in addition, can command those same creatures. These abilities, all products of death magic, are pretty much unique to the death knight alone, whereas there are plenty of powers that are shared between multiple classes (IE, the light being used by pallies and priests, nature/animal magic being used by druids and hunters, etc.)
From a gameplay perspective, death knights have a unique resource system that, to this day, is set fairly well apart from most of the other resources in the game. Combine that with the playstyle of not having a lot of mobility but still having a lot of ways to close on your opponents and ways to deal with those that like to hide out of range, making you feel like a real monster on the battlefield.
All in all, if you want to play a dark juggernaut that moves like a dark wave across the battlefield, consuming anything in your path and growing stronger for it, then the death knight is for you.
If you’re still not feeling it, I’d recommend levelling another DK through the Legion Class Hall campaign. Class-based content is always great to get in the feel for any class, and I’d say that DK has one of the better order hall campaigns.