Karthus is a great champion - one of my all time favorites. However, right now, Karthus **has a slight misalignment between his gameplay and lore identities.**
Karthus' gameplay identity is that of a battlemage. He scours nearby enemies with Defile and Lay Waste, preventing their escape with his Wall of Pain. He prefers to die in the middle of the fight so that his passive, Death Defied, can allow him to continue to deal damage to enemies. He is at his most powerful in the center of a conflict.
Karthus' lore identity is slightly more complicated to explain, and is steeped in hundreds of years of mythology. Like Mordekaiser, he is based off of WoW's Arthas (the Lich King), but Arthas' mythology is based off of the first Lich, an eastern European legend of a man called Koschei the Deathless. Koschei was said to be a powerful, skeletal warlock who kept his death inside of an egg, inside of a duck, inside of a rabbit, inside of a hare, inside of a goat, inside of a chest, on an island in the middle of the ocean, surrounded by his undead minions. Some of you who enjoy Harry Potter may also see connections to the Tale of Beedle the Bard called "The Warlock's Hairy Heart," which is based on the Koschei legend. (Rowling borrows heavily from Eastern European mythology in general, and it's a setting for a lot of the stories that went untold in the series proper.)
In Riot's official lore, Karthus is more of a wraith than a true Lich. The crucial difference is that he has no death; Karthus is not undead because of his own power or any kind of magical immortality ritual, but because of the power of the Isles. However, he still carries many of the connotations of Lich-hood. One important detail is that Liches are necromancers who can raise and command undead. Karthus is fascinated by the Black Mist and by the power and clarity granted by undeath. In his lore story, he attacks a Bilgewater ship and ~~kills~~ frees the spirits of the entire crew, resurrecting them as wraiths to join his ~~horrific~~ joyful undead chorus.
In Karthus' mind, undeath is a blessing. Attacking and killing enemies is joyous - a beautiful symphony of life and death. Why would he not give this gift to everyone?
Passive - Death Defied: Unchanged
Q - Lay Waste: Unchanged.
W - ~~Wall of Pain~~ **Mortal Divide:** Now applies Deathsinger's Thrall to enemy Champions or Dragons who pass through the wall. Deathsinger's Thrall lasts for 10 seconds, and its duration is refreshed whenever Karthus deals damage to a unit.
If an enemy Champion or Dragon dies under the effects of Deathsinger's Thrall, its spirit will be revived as a Wraith (see also: Children of the Grave). Wraiths are controllable through Alt-Click, but when left alone, will prioritize enemy Champions affected by Deathsinger's Thrall. Wraiths have a static 1.0 attack speed, and deal damage on each auto attack based on Karthus' AP. They also gain HP equal to Karthus' bonus HP.
If Mordekaiser and Karthus both deal damage to a Dragon, the champion who dealt damage *last* will control the Dragon.
E - Defile: Now applies a stacking debuff that slows enemy movement. No longer needs to be toggled off to receive the mana regen for killing enemies.
R - Requiem: Applies Deathsinger's Thrall as soon as Karthus casts the ability.