My first suggestion would be to reconsider the idea of playing a Death Knight. Mechanically they are primarily a melee class, but in roleplaying you have the freedom to cast magic beyond the limits of your in-game spellbook. The reason I say this is because by playing as unholy, you could have a ghoul companion which you can control the movement and behavior of, and because this would be the only way to have a physical humanoid minion (as opposed to something more ghostly or beastly, which are also cool but not quite as iconic for a necromancer). Neither of these perks are required, but I think they deserve consideration!
The idea of playing a warlock and using the voidwalker as a wraith would have that first perk for controlling your minion. You would need to explain to others that it is not a demon of course, but it's still a very neat idea.
Using a non-combat companion and playing a mage or shadow priest is a clever idea, as long as you don't mind carrying the pet biscuits. It also has its own appeal, because a necromancer (different from the Ebon Knights) will not want to attract too much attention. Limiting his reanimation to an individual hand, a stitched hound, or a fungal creature would all be effective ways to lay low... with the two ghost pets being slightly less safe options. You could consider the idea that your character is calling and communicating with the wandering undead spirits, rather than ripping them from their resting place and enslaving them.
It's up to you, and should be an interesting result either way!
A Lich is just an undead spellcaster. So a Forsaken mage or warlock is technically a Lich.
According to the the same Wowpedia page you linked, they are sorcerers possessing -tremendous- magical power, and were first created by Kil'jaeden from orc sorcerers after Draenor's destruction. They can cast necromancy and frost magic.
Though it isn't explicitly said, I would assume the ritual of becoming a lich is a separate level from how other undead, like zombies, ghosts, and skeletons, can be risen and still cast magic.
Personally - I don't understand how the transformation into a Lich causes the person's skeleton to abnormally grow larger and sprout tusks.
The answer is probably a combination of cool art design and the ascendancy to a lich being a special gift. The Wowpedia article actually said that the orcs were given "new undead bodies" and were "transformed into twisted abberations." If the art team was actually trying very hard to complement the lore, they might have added made the tusks a universal trait for liches since it would otherwise take two models, one for humans and one for orcs, for this to make sense.