The best choice for starting items is probably boots of speed health pots and a ward.

Ok, akali adc may seem weird and since she is melee, you'd think this won't work. I tested it multiple times, it does work. There is almost nothing she can do untill level 6 though since most of the adc champions are ranged. She can though throw her Q and farm while her support eventually poke them, or even stuns/pulls them so you can attack them. When you get level 6 you can actually engage, and that's when the fun starts. Adc champions are usually the most squishy champions. This makes it easy for akali to kill them. Late game, when you are (almost) full build, the adc won't be able to kill you anymore, you should be the on to immediatly engage and akali is a champion who can easily take down a squishy adc, this makes her a great adc in my opinion.

The runes and masteries I put here may seem strange or even very strange. Yeah, they are strange, I know. But these are very good for akali. The 30/0/0 AP and AD masteries allow her to very quickly take down a target without giving them the possibility to damage her too much. The ad in these masteries combined with her passive makes sure that she can actually heal when taking down a target. Use this on the rift scuttler when you are low health to get some health back. The runes are a strange combination of AP, AD, Attack speed and crit chance+damage. The AP quints are for her early game to make sure she has some potential early-game. The other runes are... Don't question it, just do it, it works.

Don't do this in ranked until you have trained and done this various times in normal queue it's hard to play akali adc and if you don't know akali well it probably won't work.