Below are explanations for build 1, solo top/top.

If you're going to buy an item that build off a gold item, you might as well get it first for "free" gold. So you get Avarice Blade right after boots...as you're going to buy Youmuu's Ghostblade (boots first, not Brawler's Gloves, it doesn't help Akali in beginning of game). The boots them build into Ninja Tabi, it's funny, because Akali could use just about every boot there is (even Mobility...for her smokescreen, I just get Tabi because it's op early game and cheap, your choice whether you want this before or after Avarice). Then get life steal, you could either get Wriggle's Lantern (esp. if you're solo top) or Bilgewater, you want that before Hextech Revolver so you could use it's active early, and since Akali is melee, it's easy to land. Right after that is Hextech Gunblade, and we all now why, if you want, you could throw in a Long Sword and Amplify Tome early for Akali's passive. Next is Rageblade, since Akali blows a combination of spells and attacks really fast, it's a good choice, it also buffs your passive (kind of like the old Jax), the reason I get Pickaxe instead of Blasting Wand is because they both help passive, but Pickaxe costs more and it's a better "goal before recall". Sheen comes next, helping you deal a ****load of damage with your q + attack. Rylai's is optional (you could switch it for defense (i.e. Guardian Angel), but I get it for the health and make it even hard for enemies to run away (although, I only have 2 magic damage spells and w already slows, and I'm going to get Phage anyways). Next you want Trinity Force, for speed, mainly (you could also get Brutalizer instead, for cdr on your ult). Next comes Youmuu's Ghostblade, because Akali doesn't have a speed/attack speed buff, this is a good choice to make it harder for them to run away...and for you to run away (she only has one escape move). If you bought Ninja Tabi, now is a good item to trade them for Mercury Treads (and elixirs).

Below are the explanation for Build 2, jungle.