Zed is an extremely high burst damage assassin. He specializes in killing off the ADC or the AP carry. Zed is energy based so his only limitation during laning phase or jungling is his health. Zed has a lot of AOE damage and is able to re position himself with ease. His ultimate is an amazing initiation and allows zed to almost instantly kill his opponent.

Zed's runes are flat AD marks and quintessences. Zed's ability living shadow provides 25% more bonus AD which means flat AD or AD per level is more effective on Zed then armor penetration. His Seals are flat armor which allows for an easier jungle or if you are solo top it provides additional defense. For glyph's I suggest flat magic resist or magic resist per level. Zed can also take Mid lane and having flat magic resist gives a huge advantage over the enemy mid.

Zed uses 21/9/0 masteries. These masteries are chosen so he outputs the maximum amount of damage. By taking defense masteries such as extra armor and MR he is able jungle more and lane more effectively. The 21 in offense allows for easier last hits also Zed's passive works on minions allows for simple last hits.

Since Zed is an AD assassin the simplest and easiest way to maximize his damage output is to stack Bloodthirsters. Since Zed's ability living shadow provides 25% bonus AD Zed can reach 594 AD by using 3 bloodthirsters a frozen mallet and a last whisper for armor pen. Also, Zed's boots depend on who he is facing lane if he is facing an AP buy mercury's treads if facing an AD or jungling buy ninja tabi. Although if doing well in lane and extra armor or magic resist isn't required purchase boots of swiftness or mobility. Buy a forzen mallet for extra health it also stops enemies from escaping. If Zed is unable to snowball refer to build 2 you will still output high damage and you will be tankier and not at risk of having to re stack bloodthirsters if you die.

Zed's passive contempt for the weak deals 8% of the targets health in magic damage if the target is below 50% health. This allows Zed to last hit easier and be able to deal significant damage to champions who stack armor such as Malphite or Rammus (I still don't suggest trying to kill them though :P).
Zed's Q Razor Shuriken's is a high damage long range skill shot. It passes through enemies although it deals more damage to the first target hit.
Zed's W living shadow provides him 25% bonus AD at max level. This ability allows Zed to reposition himself as well as to use his abilities at an extremely long range. When you use Zed's Q with living shadow active the shadow and Zed will toss shuriken's although the extra shuriken's will deal 50% damage.
Zed's E is an AOE slow that only hits targets around him. It deals decent damage and if living shadow and Zed both hit there target the slow will be greatly intensified. Zed's Ultimate is Death Mark Zed dashes to a target and spawns a shadow behind them this shadow will act as a second living shadow allowing you to have a max of 3 shadow clones at once. Death mark will also leave a mark that will detonate 3 seconds after it is placed it will deal 50% of damage dealt by Zed to the target as well as all of his attack damage.

Zed is requires Smite when he jungles.
He is also good with ignite to finish off kills that would otherwise get away.
Exhaust is also liable to stop targets from fleeing from his ultimate.
Flash can be used as well in order to initiate or escape with ease.

Zed has high burst damage and is able to take out an enemy carry with ease as well a being highly mobile. He can play in any solo lane position effectively and is able to win most lanes.