Since it usually takes weeks before the best build for a champion is discovered and popularized, I won't go into too much detail on the builds yet. These are just prototype builds that I've had success with on PBE and are subject to change (probably not drastic, likely just item order or swapping a few items out for others).

Like most melee assassins, Akali runs Domination primary with Electrocute. She has some options for her secondary but I'd recommend Resolve almost every game. Although precision (for Triumph) and sorcery (for more damage) are tempting, Resolve secondary has been way too strong for melees all season. It's good into ranged matchups because Bone Plating helps you survive poke and it's good into melee because the extra health from Bone Plating and Chrysalis are much stronger in early fights than any damage rune from Sorcery or Precision.

Electrocute - A must run for Akali. Very easy to proc in any combo and will increase your burst to secure kills throughout the game. There isn't really an alternative keystone for burst assassins currently.
Sudden Impact - Again, a must run rune. Procs through both your E and R. Since this rune grants both lethality and magic pen, the fact that Akali has a mixed damage kit allows her to easily abuse this rune.
Eyeball Collection - Honestly, Eyeball Collection, Zombie Ward, and Ghost Poro are all viable (Zombie ward only if you're going to buy sweeper or you build duskblade for some reason). Basically, pick the one you prefer (although I'd default to Eyeball Collection if you aren't sure).
Ravenous Hunter - Again, I think Ravenous Hunter, Ingenious Hunter, and Relentless Hunter are all viable. Run Ravenous if you want sustain in teamfights (and perhaps a tiny boost to help you with those montage 1v5 plays), Ingenious if you want your Hextech Gunblade and Zhonya's Hourglass to be up more often (these are very important cooldowns so don't count this rune out just because it doesn't give combat stats!), and Relentless if you want free mobi boots by midgame for better map rotations and faster flanking. Since I personally love mobility, I'd personally recommend defaulting to Relentless if you aren't sure, but try them all out and see which you like best.
Bone Plating - Essentially an additional 200-300 health for your laning phase. Great against poke and helps you win 1v1s or get the health advantage vs melee champions in lane.
Chrysalis - Run this versus most non-poke matchups. 50 hp gives you a huge advantage in early laning and fights. By the time the 50 hp isn't as significant, the rune swaps to extra damage. Versus poke matchups, consider running Second Wind instead for a mini Doran's Shield passive to help you survive lane.

Flash - This is a MUST HAVE for Akali. If you don't run this spell, you will die to ganks, miss out on kills, and you'll get flamed by your teammates. Also lets you pull off cool combos like R-F and Q-F.

Ignite - If you're playing mid lane, you'll almost always want to run ignite for the additional kill pressure to snowball the game. Since mid lane is pretty short, you usually won't need TP to get back to lane.

Teleport - You'll usually want to run this top lane to get back to lane after a back / death, to roam bot, and to splitpush or clear waves without losing baron. Ignite is also viable top but you have to play very carefully. If you run ignite, the enemy jungler will likely camp you simply because of how hard it is to catch up if you die with ignite top (takes a lot longer to run from base to top than base to mid). However, the additional kill pressure can snowball the game, so run whichever one you prefer.

Starting Items
Doran's Shield

A good starting item for almost every lane as Akali is an energy based champion that doesn't need mana (so Corrupting Potion isn't very useful on her). The lane sustain is great for both poke lanes and sustaining through trades in melee matchups.
Long Sword

If you're playing a matchup where you want to go for an early first blood or just feel like you don't need the extra sustain from Doran's Shield, you can also start Long Sword (plus refillable if it's an easy lane and 3 pots if you need the extra health). Starting Long Sword accelerates your Hextech Gunblade power spike and deals more damage than Doran's Shield in lane.
Early Game
Hextech Revolver

On first back, you want to start building Hextech Revolver unless you plan on buying a Hexdrinker (see Situational Items). Once this item is complete, it will be your first big power spike in lane.
Bilgewater Cutlass

After Revolver, buy a Cutlass to build into Hextech Gunblade. The active helps get you in range for kills and the extra AD will still be useful as all of your abilities have both AD and AP scaling.
Boots of Speed

Generally, buy this item whenever you have the gold (for example, not enough gold for another Amplifying Tome or Long Sword).
Core Items
For Akali's core items, there's one must-rush item (Gunblade) and 3 items you can build in any order you want. Generally, you'll want to buy Gunblade into Hourglass when fed (you want survivability over extra damage when ahead) and Gunblade, Lich Bane, and Morello when behind (you need extra damage over survivability when behind).
Hextech Gunblade

This item should be rushed on Akali for numerous reasons. First, it gives her both AD and AP, both of which are useful for Akali. Second, the active is really strong and can help you hit both your E and R. Third, Gunblade's passive allows Akali to heal from her burst, helping her stay alive in a fight. Finally, both Revolver and Cutlass are excellent item components in lane, meaning you are still hitting minor power spikes as you complete the components this item.
Zhonya's Hourglass

Buy this as your second item when ahead (as you won't really need the additional damage from Lich Bane when ahead) and third if you bought Lich Bane. This item is fantastic on Akali because your role in a teamfight is to dive the backline, kill a squishy target, and not be blown up in the process. The hard part isn't exactly killing the enemy squishy. Rather, it's surviving after your initial burst. Being able to kill the enemy ADC means nothing if you simply die right afterwards. Hourglass active helps to solve this by keeping you alive long enough for the enemy to have to focus your team instead.
Lich Bane

Lich Bane's passive synergizes well with Akali's kit as she has to proc passive multiple times in a fight (your passive autos also proc Lich Bane). When buying this item, you'll generally want to buy Sheen first if you can build the whole item. If not, build Aether Wisp > Blasting Wand > Amp Tome (in that priority). Don't worry about the extra mana from Sheen as Lich Bane is still cost-efficient on Akali without the mana.
Morellonomicon

Lots of damage (AP + mpen), grevious wounds, and some tankiness to boot. Buy this for extra burst versus enemy squishies and rush it as a second item if the enemy team has a healer you want to shut down (e.g. Dr. Mundo or Soraka). Your build order for this item should be Oblivion Orb > Blasting Wand > Oblivion Components.
Situational items
Rabadon's Deathcap

If you're fed and want even more damage, Deathcap is an excellent option. It literally does nothing but give you more damage.
Void Staff

Build this if the enemy team is stacking MR or if you're behind and want extra burst along with Lich Bane and Morello.
Guardian Angel

Buy this when you're really far ahead and the enemy team's win condition in a fight is to burst you down. This passive basically denies the enemy's win condition as Akali has lots of healing and options with W once she revives, meaning it's hard for the enemy team to simply kill you again once you revive (unless you suicided 1v5 :P).
Spirit Visage

If you're looking for a tankier option to survive enemy magic damage, Spirit Visage is a good option. You get a lot of healing from Gunblade's passive and Q (plus Ravenous Hunter if you ran it) so this will keep you very healthy in fights.
Hexdrinker

A situational item that you usually buy as your first item and leave in your inventory. Buy this against heavy AP burst such as Syndra or Fizz so they can't 1-shot you once they hit 6.
Maw of Malmortius

Usually you want to sit on the hexdrinker until midgame. Then, you can upgrade to Maw if you still need the magic damage shield. The passive is quite good on Akali as she can make use of the AD, spell vamp, and lifesteal. Plus, if you combine this with Spirit Visage versus a heavy AP team, you're basically impossible to kill.
Boots
Generally, you'll want to either build defensive boots or sorcs on Akali. The choice is pretty simple. Build defensive if you want either of the passives and/or the armor/mr, and build sorcs if you want more damage.
Ninja Tabi

Buy this into heavy AD comps and / or team comps with lots of auto attackers (for example, you would want Tabis over Mercs versus AP Kaisa or Corki).
Mercury's Treads

Buy this into heavy AP comps and / or when the enemy team has a lot of CC that you want to reduce. The tenacity helps prevent being cc-locked to death.
Sorcerer's Shoes

Usually, you'll want to buy these when you get ahead early and want to snowball your lead quickly. You can also buy them if you just want extra damage.

Thank you for reading. Please let me know if you notice any outdated sections or errors and I will do my best to fix them asap.
Lastly, if you enjoyed this guide, feel free to check out my other guides!