The stealth and slow of Twilight Shroud can be used offensively to provide cover and allow her to chain skill combos on the enemy more quickly, but can also be used defensively to prevent drawing enemy focus during team fights and to provide aid in retreating after team fights or running from ganks.
Purchasing Mejai's Soulstealer early can make you very powerful late game if you are able to earn a decent number of kills without dying. Although this is extremely beneficial, it is situational, relying on your early game performance and continued success.
Without the necessary runes and masteries to activate your passive (because Akali needs her own rune page to effectively utilize her full potential), Doran's Blade is a good choice for a starting item for the added health and AD.
After Akali's burst, she has to wait for her cooldowns in order to deal her burst damage again, which takes a while, even with cooldown runes and masteries. If fighting against her, use this time to take her down.
When you are chasing her, make sure you are aware of the amount of HP you have, due to the fact that she can nuke quite easily and score another kill on you.
Against an Akali, play more offensively until she reaches level 6; try to deny her early game, as she usually has the advantage once she gets her Shadow Dance.
Akali has very few offensive items early game. Zone her out so she can not farm, which will further delay her from getting her Shadow Dance. Once she does hit 6 try to bait her into using her ability on you if you can take it, as she is less effective the less charges she has.
Twilight Shroud will stealth her only if she remains in the AoE, Meaning that she will be out of stealth when she is out of the AoE or attacking. Try to move out of the AoE as fast as possible to prevent her from bursting you while in stealth, or pick up an Oracle's Elixir or a Vision Ward if she seems to be over using it.