Okay so this build is slightly unconventional. This build, both for runes/masteries and your final items, are for a tanky version of the typical Akali build style.

Bloodthirster is the first of the two trollish items, people may underestimate you for buying this, but there are reasons it is actually powerful...
Warmog's Armor is the second of the troll items. Let's just say you will be able to live through quite a bit if you get this far into the game, and that extra life will be worth more than most other items to keep you alive against crowd control effect especially

Hextech Gunblade + Bloodthirster is another troll way to build sustain in a more bursty way. Packing Cleanse over ignite might be better for this item selection.

But remember this is a troll build! Enjoy!

I choose my Marks for AD to nearly get her passive (8.5AD). Her Quints then nearly get her other passive (14.85AP). The Seals of armor and Glyphs of MR make her resilient into mid and later game allowing safer building of burst damage items.

Also, depending on your play style you can swap out Greater Seal of Armor for Greater Seal of Defense Which I may try whenever I have IP to spare on more runes.

Side note: I've gone Legendary 3 times after owning her for only 2 days, using this build.

Although there are other Mastery styles out there, I made my tree this was for use with my more favorite Summoner spells. I take 3 points into attack damage giving me 11.55 total and granting me her first passive. I take the 4 points of ability power adding to 19, gaining more from the Blast and Archmage granting me 20.84 AP for her second passive. I take as much offensive as possible, but put the remaining 5 points into Ar and Mr focusing the extra MR specific for sustaining in middle lane.

Optional: If you prefer to just have 10.55 attack damage you can swap 1 point out of Brute Force for a point into Summoner's Resolve or Hardiness

Item order is not super important, but the items in this build are all extremely important.

Starting Items:
Boots: An early gank mid or top can be devastating, grabbing boots first make you less vulnerable and 3x Health Potions to stay in lane

On return to base:
Bilgewater Cutlass: I've had really good success with rushing this item. I gives you some extra burst early game as well as slow and lifesteal. Because it is AD it also increases spell vamp to about 16% or so.
Rylai's Crystal Scepter: Grants early AP, Life, and Slows target for epic chase downs
Rabadon's Deathcap: Grants superb AP bonus to melee attacks (+30% damage as magic) as well as buffing her overall damage, if the enemy is playing passive or you don't feel you need gunblade for sustain or Rylai's for life/slow, grab an early Needlessly Large Rod

End Game:
Lich Bane: Filling in with lichbane procs a 385 bonus to physical damage making Q and R specifically more potent, but is heavy investment if bought too early
Bloodthirster: Grants Spell Vamp via Akali's Passive, life steal, and will add 60 damage to your E. (Rabadon's Adds 54)
Warmog's Armor: Finishing off With Guardian Angel seems to be common, but I'm not a fan of the item in general and would rather just have more life or more Damage from an AP item. The Giant's Belt can be bought for mid game sustain and converted later if need be

I like the illusion of being weak (as most Akali play damage heavy with runes and masteries) but that extra sustain will allow you to be not just an assassin but also a strong team player late game. I generally try to rush Rylai's or Rabadon's.

Situational:
Black Cleaver or Hextech Gunblade can be swapped for Bloodthirster but I prefer the BT because of its damage bonuses
Zhonya's Hourglass Void Staff or Hextech Gunblade are all acceptable to replace Warmog's Armor with
Of course finish off with Elixers.

*Note if Warmogs is replaced by Gunblade you gain incredible sustain if the opposing team cannot place crowd control effects on you. Warmogs is best for surviving late game with CC, nukes, and ulti's flying, but if bursty is your style, then grab Bloodthirster LAST and Gunblade earlier in game.

Pros:
Safer play with defensive options
Strong early game with double passive
People won't be mad about you lacking passives
Strong late game
Stronger team play, tankier in general, esp with Warmogs
Strong mid or top lane
Strong sustain with dual leech, esp if going BT

Cons:
Slightly less AP damage than other builds
Slower attack speed (without trinity or rageblade as some people get)
Deals a bit less damage if going warmog's
Deals mixed damage can make you weaker vs some targets

The big thing about Akali is that she is an Energy user (Ninja). This allows her to stay in lane longer as long as she has life. The double passive is a big deal for both Spell Vamp and Magic damage on strike.

Using Q on a creep to kill it is a caution. Because Q grants her energy on striking the Marked target you always want to follow it up with a basic attack if possible. It's not overly bad, but it can leave you low on energy if you do this often, leaving you more vulnerable to a gank or a counter-attack. Q can be used to harass enemy champions, and once you have your ultimate you can Q and wait a second for perfect positioning to allow Q to be ready again and to regain a little energy, then Dance onto a target, consume the mark instantly, and Q+E for a quick 2nd Mark and Consume and E to potentially finish off your target.

Remember that you only have 3 Essences (leap attacks), these can be used to jump to neutral creeps or backwards into a line of creeps to dodge a gank, a failed assassination, or even a successful tower dive.

Final Stats:
222 Total AD (111AD + 111aD)
430 Total AP (318.5 * 1.35%)
168 Bonus Magic Damage (76%)
430 AD Lichbane Proc
25% Total Spell Vamp
20% Total Life Steal
3745 Life

This stats are with max stacks of both Warmog's Armor and The Bloodthirster.
It does not include Elixirs, Baron Buff, or Lizard Buff

Gunblade Vs Bloodthirster
Gunblade grants 15% more spell vamp to this build and 5% less life steal.
Attack damage falls by 55, AP jumps by 65
Magic damage dealt on-hit is -15 even though it is a higher proportion (91%)
Therefore all basic attacks drop by 70 damage total.
Lichbane procs are increased by 87 with Gunblade
Gunblade also has a 300 magic damage nuke
Crescent Slash deals 13.5 less damage with Gunblade, but Akali's other two spells gain 35/35(Q) and 43(R) magic damage

Getting both for a pure burst build yields +87AP +45AD, 52% SV, 35% LS, 267 basic, 243 magic on-hit(+80), 517 lichbane procs, but have 2,475 Life

(Items in the above math are boots, RDC, LB, RCS, BT, WA)

Gunblade + Bloodthirster Vs Warmog's
+145 AD
+65 AP
+300 Nuke
-1380 HP
+35 life steal
15+24=39% spell vamp