Akali is a very mobile, very bursty assassin character. Despite what anybody tells you, she CAN carry and this build is meant to help you do that. When I get to the items chapter I will explain further why I buy each item, and why I buy them in the sequence that I do.

These runes (combined with your masteries) will help you take advantage of Akali's glorious passive. 8% spell vamp at level 1 is a very good thing to have, the extra damage on your auto attacks will also help with last hitting, should you be playing against a passive laner or melee champion.

Once more, these are mainly to take advantage of your passive, combined with your runes you can do this easily.

I always start with boots on Akali, cause I play her mid and combined with her shroud, boots, flash, and the short distance between the middle of the lane and your tower, it will make many escapes from ganks easy. Keep in mind, this doesn't make you invulnerable to ganks, and you still want to play cautiously. Next up, I grab a hextech revolver for the spell vamp. 20% spell vamp after your first trip back is nothing to laugh at. Followed up by a rylais crystal scepter, to give you that slow (which makes chasing people down with your ultimate even easier) as well as the hp to help make you a little bulkier. After that I finish my gunblade, because now is the point where you can easily burst down almost anyone, since they cant escape very well with the slow from your rylais, the slow from your gunblade, and the fact that you can use your ultimate 3 times (all 3 charges). Next up, deathcap, standard ap carry item and isn't too surprising. Also make sure you upgrade your boots whenever you feel it's necessary into whichever pair will work better for the current game.
Once you're around this point in the game, you'll probably be a big target, so a GA will make sure you get some extra tankiness plus a revive if you go down. If you can get somebody on your team to build a WOTA, get one yourself. Between the double WOTA, gunblade, and your passive, you can have 75% spell vamp. After you revive this spell vamp can get you back up to fighting material in no time. Another alternative to the WOTA is lich bane. Great item that pairs tremendously well with your q.

The skills are rather self explanatory, max q first because it's your most damage. Max e second for damage. As well as grabbing a point into w early for safety. One neat trick to do once you hit 6, throw your q on them, wait for your q to come off cd, ult on them, auto attack (or e) throw another q, then auto attack again. This will provide you with massive burst, which will be unexpected if you were harassing with your q during lane phase.

I like ignite and flash, because they help secure kills and escape. You can use other summoners if you would like to, but I would still recommend flash and ignite.