To start off, I think assassin akali is the better build when you're good on Akali. If you just started to pick her up, use the bruiser build 24/7. It will be a lot more forgiving, and you get used to the combo's etc without getting oneshot if you make a mistake. Once you get better at her you can really use Akali to her highest potential, and build assassin.

I recommend against going conqueror runes with assasin items. Akali strives when being built towards 1 direction. Let's say you play her as assassin. When you take conqueror, you lose a lot of your burts, making your job as an assassin harder. You also can't easily stack the conqueror. Stacking it requires you to hit the enemy a lot, which means they can hit you back. Assassin Akali does not have sustain, so it's a no go. Your job isn't to sustain. You need to burst the carry, and then use zhonya's/stopwatch/W/flash if you need to get out. Simply put, going conq with assassin build isn't worth it. You lose burst, and you can't stack the conq because you are squishy. Trading a lot of burst for a little bit of consistent damage(cuz you can barely stack it) isn't worth it when you need to assassinate and burst. Don't tell me "Akali has enough burst already." You get more from going electrocute than conqueror. Because while Akali may have sufficient burst, she doesn't have the sustain to stack conq without bruiser items. I'm kinda repeating myself here, but I hope you understand.

Likewise, when you go bruiser Akali, don't use electrocute. You lose a lot of consistent damage that conqueror provides. In this cases you can stack it, because you can tank a lot more hits due to your sustain. Your job now isn't to oneshot the carry (even though you most likely still can) but to be more of a frontline champ. You need to be the one to protect your carries, and fight front to back. Trading a lot of consistent damage for a little burst isn't worth it when you need to be a bruiser.

Always go sorc boots, both with bruiser and assassin, no matter the enemy team comp. You have runes to deal with cc, and items to deal with armor/mr. You NEED the 18 magic pen. Also, cdr boots is kinda cringe, as you don't have enough energy to even use the boots properly.

Don't get me started on AD akali, it is the worst way of playing her. Some people will say it's "OP" and tell you high elo Akali mains build it. First of all, it is not "OP" at all, in fact, Akali works cuz of her dashes and shroud. Without those, she's useless. What does AD or AP build have to do with this? Your abilities scale more with AP than with AD. No, no, hear me out. Yes, I know that first ult, Q and passive have more scaling on AD. HOWEVER: You can have more AP than AD + your second ult ONLY scales with AP. And here's the biggest thing: Akali has dashes. Those dashes deal damage. AP is focused on damage with abilities on Akali. AD is focused on damage with auto attacks on Akali. Akali doesn't work when you auto attack all the time. That's why she has these abilites. Akali is a melee minion with dashes when her W is down. Auto attacking all the basically makes you visible at all times, thus your W doesn't do anything for you. This makes Akali useless. Your passive auto's are the most auto's you should be using. Sometimes normal auto's, yes, but not enough to make AD items worth buying. People tell me "yeah but she has sustain with AD build." My answer is simple: Try my ap bruiser Akali build.
Elaboration on the scaling: would you rather have 300AD with 65% scaling, or 600AP with 60% scaling? Yeah, I think the answer is clear.
Bottom line: AP fits Akali's kit better than AD, Akali deals more damage with AP than with AD.
(I think I got started, yeah.)