There exists an ancient order originating in the Ionian Isles dedicated to the preservation of balance. Order, chaos, light, darkness -- all things must exist in perfect harmony for such is the way of the universe. This order is known as the Kinkou and it employs a triumvirate of shadow warriors to uphold its causes in the world. Akali is one of these shadow warriors, entrusted with the sacred duty of Pruning the Tree - eliminating those who threaten the equilibrium of Valoran.
A prodigal martial artist, Akali began training with her mother as soon as she could make a fist. Her mother's discipline was relentless and unforgiving, but predicated on the fundamental principle: "We do that which must be done." When the Kinkou inducted her into the order at the age of fourteen, she could slice a dangling chain with a chop of her hand. There was no question - she would succeed her mother as the Fist of Shadow. She has had to do much in this role which others might find morally questionable, but to her it is in service of her mother's inviolable doctrine. She now works with her fellows Shen and Kennen to enforce the balance of Valoran. This hallowed pursuit has unsurprisingly led the triumvirate to the Fields of Justice.
"The Fist of Shadow strikes from the cover of death itself. Do not impede the balance."
Akali is a hybrid anti-carry champion with huge burst damage and is a fantastic chaser. She is probably the best hybrid character in the game with her passive, Twice Disciplines, which adds magic damage to her auto attacks based on her AP and spell vamp based on her AD. She has Mark of the Assassin for burst damage, Crescent Slash (which deals AD) and AOE damage, Twilight Shroud for defense, and Shadow Dance for chasing and juking.
The biggest drawback for Akali is that she is squishy AND melee. She alleviates some of this while hiding in her Twilight Shroud, but it has a fairly long cooldown for how often you are going to want to use it. But the best way to prevent your death is to just do what Akali does best; run in, assassinate someone, and run out (Twilight Shroud/Shadow Dance for safe exits), while your tank takes all the hits.
While there are several champions that can very viably go Hybrid (Shaco, Twisted Fate, Ezreal, Kog’Maw, etc), most of these characters would actually do greater damage by just picking AP OR AD; whereas Akali would be missing out by only picking one. You are losing either extra spell vamp or extra magic damage, both of which are going to make you far more deadly if you have them.