This build is optimized for ARAM, not Summoner's Rift.
You constantly play around with Snowball Summoner Spell, either by marking your smoke with E after you hit the snow ball, use the second activation to go, use Q and return to your smoke or by abusing the snowball mixing it between your combos together with Rocketbelt making you almost impossible to hit mid combat.

Warmog's is absolutely necessary since it allows you to be an ever present threat on the bridge. As long as they don't kill you, you aren't really out of combat and can always return full health, and your full health is a big health pool.

You can lash out a LOT of damage to the point that squishy targets will do everything to avoid you, but even more tanky targets can't ignore your damage.

Lichbane will basically just double your passive, while also giving you some extra movement to get out again and trigger your Warmog's.

Cosmic Drive will just double down on that, to give you even more mobility for you to come in hot and get out even hotter.
Constantly going in and out of the fight, you won't just break your enemies towers, but also their very will to fight.

When there is someone you can't really get that can easily kill you but just won't die even when you actually hit them a lot and always returns at full health, it can cause a lot of despair.

If the enemy team has a Zed, Karthus or other champion with an ult that is virtually impossible to dodge, you should swap Lichbane or Cosmic Drive for Zhonya's Hourglass. You can still use the snowball on a minion and use the second activation with proper timing instead of that, though.

If you need anti-healing, pick Morellonomicon in exchange for one of your last items. You can make the Oblivion Orb early, but only finish Morello as either your 4th or last item.

Another alternative is the Banshee's Veil, depending on who the enemy champions are. but this one is very situational when the enemies don't have much poke and your issue is only one or two skills they use to engage on you or to prevent your own engage.

If you need more magic resistance, either pick Spirit Visage or Force of Nature, but if the enemy team has 4 or all 5 as magic damage dealers, pick both as your last 2 items.

Force of Nature is your main choice of Magic Resistance, since it grants both a high value of MR and a lot of movement speed while cutting down all magic damage by 25% flat.
7 AP on Demonic Embrace.

Spirit Visage on the other hand is really good if your team has a lot of healing or barriers, like a Soraka, Janna, Lulu and similar champions to support you. It will give you a good amount of Ability Haste.
9 AP on Demonic Embrace.

If the enemy team is mostly physical damage, though, your options are between Dead Man's Plate and Randuin's Omen or both, if they have 4 or all 5 as physical damage dealers.

If you only need enough armor to get in, kill the priority target and get out, pick Dead Man's Plate. It is surprisingly synergetic with this Akali build, since the distance covered by your E and by the snowball counts towards the charged effect, so every time you use your second E activation or second Snowball activation is gonna have the extra explosive damage effect with slow.
6 AP on Demonic Embrace

On the other hand, if you really need armor and are taking excessive physical damage, use Randuin's Omen.
It will cut down the physical damage on you by a huge margin and its active is extremely useful on your second E activation and to prevent enemies from escaping while you trigger your passive empowered strike.
8 AP on Demonic Embrace

Lastly, if they are mostly tanks and building a considerable amount of MR, you either want Void Staff, Shadowflame or both if only one is not enough.

If your own team is mostly physical damage and the tanks don't have too much MR or if they have a lot of barriers, build Shadowflame as your 4th item.
4 extra AP on Demonic Embrace.

If your team is mostly magical damage and they are building a lot of MR, build Void Staff. If needed, build both and Sorcerer's Shoes.

SnowPoke:
Abuse your Snowball as much as possible.
If you hit the snowball, use your smoke and tag it on a safe place with your E, use the second activation of Snowball, use your Q on arrival on the biggest number of enemies you can hit and then use your second E activation to get out.

You can also use your Snowball to mark an enemy minion near your team when you hit your first E activation on an enemy.
Go with E2, hit AA, Q and get out on the second activation of Snowball on that minion. Just be careful to hit a minion that is full health and that your team don't kill that minion.

Other than that, it is business like usual for any Akali player.
Prioritize squishy targets, remove threats, wait for your team's tank to engage first if possible and make the enemy team spend their resources and return to safety to heal before the next engage, which Warmog's is gonna make sure you are always at peak condition.

Your presence by itself will wear down the enemy team and if they can't see you, they will be even more stressed out, avoiding to start a combat until they know where you are at the risk of you engaging their backline from beyond with a single snowball that hit on a caster minion.

Whenever you are below 50% health, try to abuse your smoke to get a few extra seconds in the middle of the fight. It may just be enough time to barely trigger Warmog's Armor, and then just by using your E and managing your distance a bit, you can return a lot of health before the enemies get to touch you again.