Well this is my first build ever. This is for Akali (Thats why your here I would assume). This build focus mainly on AP with a hint of AD to get the Spell Vamp part of her Passive. Ill be talking about a lot during this build so bare with me please. Also please rate and comment for anything I should add or fix. = )

PROS
-Quick to get to her target with Ult.
-Stealth Circle
-Decent AoE Ability
-Insane burst damage
-Great Chaser
-Once you get going with Akali she is a blast

CONS
-Kinda squishy
-Her stealth circle is small
-Easily focused if your doing well

Mark of the Assassin"Q"- Ok so this spell Akali throws one of her sword weapon things (haha) at the target and it does damage and marks the target. If you auto attack the marked target then you will do your auto attack damage plus additional damage that scales off of AP you will also restore some energy.

Twilight Shroud"W"- This is Akalis stealth ability. This ability simply (when standing in it) will stealth Akali and it slows enemies who are inside of it as well. Learn to use this well and where to place it because not using it can cause a death and bad placement can cause you not to escape or to be right on top of the fight. If you place this spell in a bush, you are double stealthed in a sense.

Crescent Slash"E"- This is Akalis AoE ability. This does damage based on your AD and AP combined. It doesn't have a huge range but its enough to hit all minions in a wave if you are standing in the middle of the wave.

Shadow Dance"R"- Ok this is Akali's ultimate. It is a fantastic ult. This is why Akali does not let people get away. This Ability will make Akali zoom super fast towards her target dealing damage based on your AP. This ability is on a very short cool down. Now the only catch to her ult is that you have to have Essence of Shadow Essence of Shadows (Max is up to 3). But this should not be a problem if you are doing really well because you gain charges periodically and on kills or assist.

Twin Disciplines- This is Akali's passive. What this does is whenever you get 20 ability power your auto attacks with deal 10% bonus magic damage going up by 1% every 5 AP gained thereafter. On the other hand once you get 10 addition AD you will gain 10% Spell Vamp increasing by 1% for every 5 AD thereafter.

I use:
Greater Mark of Magic Penetration x9 for some extra MP(Magic Penetration). Its pretty standard among people who use AP as a main form of damage, it simply just helps out with being able to put out some extra damage.

Greater Seal of Scaling Ability Power x9 for that extra boost of AP to help you out when you hit level six. Ill explain that later. But I use just for some added damage.

Greater Glyph of Scaling Ability Power x9 for the same reason as the Greater Seals of Force. Just looking for some added damage.

Greater Quintessence of Ability Power x3 for once again that added boost to your damage.

Now I want to fully explain my Scaling AP Runes and my Flat AP Quints. For starters I use to use Flat AP runes all the way around (Yellows, Blues, and Quints). Then I did some math to find out if I would have more AP at level 6 with Scaling Runes or Flat Runes. It came out to be enough AP for me to want to switch (That only pertains to Yellows and Blues). Now I kept my Flat AP Quints because at level 6 the Scaling AP Runes were down by roughly 2 AP. Ill explain why I think level 6 is most important to me for choosing my runes that I did.

Ghost- I personally like to run fast and have a good escape ability.
Ignite- Extra damage for runners if you choose to use it that way, but it also can give you more AP from the mastery.

Some other good ones to choose:

Flash- Maybe you need to escape and you have your Twilight Shroud next to a wall, get in Twilight Shroud and then Flash over the wall to get away.
Exhaust - To lower your targets damage out put and help your team catch up/stay with your target.
Cleanse- Makes it really hard for you to not get to your target if your getting CCed(Crowd Controled).

Not so good choices:
Clarity - Akali doesn't use mana, no need for Clarity to restore your non-existing mana pool.
Smite- For my build it is useless, but if you are a Jungle Akali then this would be useful.

I simply just liked the extra damage out put. Also having the extra CDR(CoolDown Reduction) and MP is nice as well. I also liked getting that little bit of additional damage from the Offensive tree since I only get one AD item in my build. I also got Improved Ignite for the extra AP when it is used. I got the AS(Attack Speed) mastery because sometimes its easy to dash at your target, mark them but not get an auto attack off for your true burst. This just helps give you a better chance to get your auto attack off on your target when they are marked. As far as taking more Crit(Critical Strike) Damage versus more Armor Pen(Penetration) is simply up to you. Armor Pen is very viable seeing that Akali is a Melee Champion. So its just all of a matter of what you prefer.

I start with a Tome for the AP, that and it builds into Rylai's Crystal Scepter. You could start with a Dorian's Ring for 5 less AP but 100 HP(Health Points) and get some useless Mana Regen, all up to you. By the time I have to go back my first time I like to have at least 1100g(Gold) in order to buy my boots which help with getting even more MP for some more damage. After that I got back to get another Tome and a Blasting Wand. I get another Tome so that I can build that into a Mejai's Soulstealer after I finish building my Rylai's. Now if I am doing really well I may get my Mejai's before finishing my Rylai's but if I'm just doing bad or alright I get my Raylai's first for extra survivability and the passive slow that comes from it. This slow really helps you stay on top of your enemy, between your ult (Which procs Rylai's passive) and Rylai's passive your target should never escape their death hehehe.
Now after Rylai's and Mejai's I go into get a Sheen which give extra AP but the main reason for it, is its passive. That extra damage from your auto attacks (with Sheen) plus all the other damage that you get from your auto attacks ("Q" and Akali's Passive) adds up to make some amazing burst damage. So after Sheen is when I finally get my first lick of AD to get the extra SV(Spell Vamp) from Akali's passive. So I get a Pickaxe and build it up to a Bilgewater Cutlass ASAP. This will give you the AD/SV part of Akali's passive along with a nice active passive from the item and some LS(Life Steal). After I build that I will grab another Blasting Wand and then build my Sheen and Blasting Wand into a Lich Bane for some MR and more AP. Yes I know the item has Mana and Akali doesn't use mana but the passive on Lich Bane is just to good to pass up on Akali. Now after Lich Bane I will start to build a Hextech Revolver, from there I would combine the Hextech Revolver and the Bilgewater Cutlass into a Hextech Gunblade. After that I would buy a Needlessly Large Rod and then build that into a Zhonya's Hourglass. You could build a Rabadon's Deathcap instead if you prefer but I prefer the extra survivability that Zhonya's Hourglass provides.

This section I'm just using for nifty facts about Akali or things to try with Akali to help you survive and get kills.

1. You can ult through terrain. Basically...if you can see then and they are in range of your ult, you can get to them.

2. Placing your Twilight Shroud partly on a bush will let you go into the bush stealthed to see if anyone is in the bush.

3. Placing your Twilight Shroud at an intersection in your jungle or near multiple bushes then this allows you many choices of direction to escape from your enemies if need be.

4. Lead your target on by placing down TS(Twilight Shroud)(not needed all the time) and using your "Q""R""E" order on them and then run away or into your TS wait for your enemy to be stupid enough to come back since you got "scared" and ran, then when they come back do the order again. ("R" and "Q" are inter-changeable based on your situation)

5. Be careful where you place your TS during a team fight because you don't want it right on top of the battle because then well your probably taking AoE damage while among everyone. So instead try placing it just outside the fight and using your ult to get in and do some damage and then go back to TS if your taking to much damage.

6. Blue Buff (Golem Buff) is very usable on Akali or any energy using champion. Yes most people see it as for mana users for the extra mana regen but Blue Buff also gives you energy regen and more CDR. So if your running into energy problems go grab Blue Buff.

7. Red Buff (Lizard Buff) is great as well because there is even more slowing power if your Rylai's wasn't enough and its does EVEN MORE AUTO ATTACK DAMAGE...that is now Lich Bane, Akali's Passive, Mark of the Assassin, and Red Buff?!?! Oh man haha thats a whole lot of damage.

8. Place Ignite on your target at the start of the battle so that you get that extra AP from your mastery, you shouldn't need it for escapees (You can use it for that reason not saying that you can't just how I use it) because you have your amazing ult to get to them as they are running.