WHAT MAKES HER STRONG?

Shadow Dance with maximum 3 times gives her extreme mobility to chase down enemies.

Mark of the Assassin deals heavy burst damage by 260+0.9*AP in total.

Twilight Shroud makes her invisible and more tanky.

Hybrid damage dealer

Twin Disciplines might be one of the most special passive.
This makes every basic attack dealing bonus magical damage by AD*(AP/6+6)/100. For example, if you have 100 AP, this increases the damage of your basic attack by 23%, 200 AP for 39%, 300 AP for 56%, 400 AP for 72%, 500 AP for 89%.
Additionally, this provides you spell vamp by (bonusAD/6+6)%. For example, if you have 50 bonus AD, it gives you 14% spell vamp, 100 bonus AD for 23% spell vamp, 150 bonus AD for 31% spell vamp, 200 bonus AD for 39% spell vamp.
This passive can give you both sustainability and damage.

Ignite deals some nice damage. If you are determined to kill, this would be the best choice.

If you want to assist your teammate or pick a free kill, bring Teleport.

If your opponent has deadly killing ability in early game, Exhaust will be your savior.

I select greater mark of hybrid penetration because you need to trigger your Mark of the Assassin by basic attack or Crescent Slash in the whole game, which are both physical damage.

I pick Greater Quintessence of Attack Damage because it give you some additional spell vamp and can help you kill minions. Both are essential in early game.

Greater Glyph of Ability Power bring you some damage. If you are facing AP champion like Rumble, or Elise, you should take Greater Glyph of Magic Resist instead.

basic attack to trigger your Mark of the Assassin in early game and deal heavy damage in late game, so I pick up Spellsword and Weapon Expertise . Bonus AD also lead to bonus spell vamp and that why I take Brute Force .

Shadow Dance has shorter restoration time in higher level, that would definitely bring you dominance.

Mark of the Assassin will be your main damage skill all the game, so maximize it without second thought.

I maximize Twilight Shroud second in terms to increasing my sustainability. It gives you not only invisibility alsobonus armor and slowing effect, which will be more useful than higher Crescent Slash in team fight.

STARTING ITEMS Health Potion SIGHT WARD
It's harder for Akali in early game since the amount of red pots have been restricted to 5. Unfortunately, you don't have better options to survive early game. If you are facing an AD champion, take Cloth Armor first.

EARLY ITEMS
Try to obtain Bilgewater Cutlass in the first time you recall. It's better than Hextech Revolver because the cutlass gives you life steal and also some spell vamp from your Twin Disciplines while the revolver provides only spell vamp. The passive of the cutlass, additional damage and slowing effect, is also useful for kill. Afterward, take Seeker's Armguard or Negatron Cloak according to your opponent.

Mercury's Treads
In team fight, I think it's more disappointed being focused than letting enemy escape with low HP. The former might make you instantly dead. Reduce the effect of control skills, and then you will have more chance to kill.

Hextech Gunblade
This boosts your spell vamp by its passive and also your Twin Disciplines. You may have at least 33.5% spell vamp after you purchase it. With the defense from your Twilight Shroud, the HP you drain will be more valuable. If your opponent is willing to duel with you, don't miss this.

Zhonya's Hourglass
This item give you both damage and armor. More important, as long as you are focused in team fight, the active effect will save you.

Spirit Visage
An incredibly cheap item. The bonus healing fits her well after you purchase Hextech Gunblade and CDR can make Essence of Shadow restore faster. If you are facing an AP champion, you should take this a bit earlier. Its only shortcoming is that 200 HP is not really much, so you may consider to get more HP or Guardian Angel after this.

Rylai's Crystal Scepter
It gives you HP, AP, and also a slowing effect to chase fleeing enemy. A great hunting weapon.

Lich Bane
This significantly increases your burst damage since you always need to trigger your Mark of the Assassin. Even that 250 mana are wasted, I still consider it as a wild item for her.

Your main damage source is Mark of the Assassin, which has 260+0.9AP magical damage in total and also bonus damage from Lich Bane and Twin Disciplines if you use basic attack to trigger it. Thus your goal is doing Mark of the Assassin and basic attack as much as possible.

MOST POWERFUL COMBO
Q on your target -> W around him/her and chasing him/her for Lich Bane nearly CD
-> R to your target then A+E+A+Q+R+A.

BASIC COMBOS
Q+R+A+E
R+A+Q+A+E

Practice it and you will find out your target die in a second.