So I bought this lovely champion Akali and I was going to get her a skin, but I wanted to test her out first. And guess what, I suck (I bet that doesn't surprise you...) So, I need the wisdom of you summoners to help me.
(ﾉ◕ヮ◕)ﾉ*:･ﾟ✧ Shower me with your wisdom!
This is how I build her. Those hast two items can be swapped out with others, but you NEED Lich Bane and Gunblade. Hope this helps. :}
Gunblade may be the ideal item on Akali because of her hybrid kit and passive. The attack damage gives her spell vamp on top of the lifesteal and spell vamp of the item itself, which is important for durability, and the ability power is, well, ability power. The active is also good for locking down targets, while other 15% health actives (like 
) aren't as useful because you should already have done a good bit of magic damage to your target before you use it.
If you scale with AP, you want Rabadon's. Pretty simple choice.
Zhonya's is often a good choice on Akali. The ability power and armor is always nice, and the active is good for surviving burst damage, delaying for your cooldowns, and just staying alive.
or 
Sorc shoes help with dealing damage on Akali, but I find myself buying Merc treads instead quite often. If an assassin gets CCed, they need all the help they can get or they're sunk, and Akali, not having a hard escape (there are a few advanced tactics using her shroud and ulti, but they're situational) is no exception.
I tend to alter the rest of my build a bit based on what defensive items the other team is building. If the team is building mostly armor, I'll build Lich Bane and Abyssal Scepter like ThatGuyNamedAce suggested, but against teams with lots of MR, I build Void Staff and start working on a 
Trinity Force. 's sheen does a little bit less damage in most cases, but after reductions for MR I think it works better than the magic damage from .
I hope that helps a little bit. Akali does take a bit of time to get used to, especially if you don't play many assassins (like me), but she can be a lot of fun and become a real threat even if you aren't having your best game. Oh, one other suggestion. If you are ahead and are able without losing too much ground in your lane, roam a little. You should win most 1v1s (don't go for their tank or someone with hard CC though), disrupting the other team quite a bit and putting your team even further ahead.
