I chose to start my build with Long Sword and two Health Potions. This should give you sufficient poke, sufficient last hit potential, and some health to survive a lane where you are at a disadvantage. If you are really scared about your lane matchup, and you are going against an AP mage, you should start with Null-Magic Mantle and two Health Potions. This suggestion is one that I give to all my mid allies, if they are in a hard matchup. For example, Annie vs Katarina. I would advise Katarina to build Null-Magic Mantle so that she can farm and trade damage effectively with Annie. If I were Annie, I would start with two Sight Wards, some Health Potions, and some Mana Potions, or get Crystalline Flask and an extra Health Potion. Annie wants to out-sustain Katarina and push as hard as she can to kill the tower and to give herself time to roam and win other lanes.
Zed is indeed an assassin, but he can be built as a strong bruiser, and the items in this build help him live long enough in a team fight to clean up afterward. The items can be replaced with stronger counterparts to them. For example, Locket of the Iron Solari can easily be replaced by Sunfire Cape or Randuin's Omen or even Spirit of the Ancient Golem, because it gives you Crowd Control reduction and a whopping amount of health. The beauty of the Locket of the Iron Solari is that it gives you that Cooldown Reduction that you so badly need. You can also use the active in the team fight effectively, because you are in the middle of the action, whereas some supports avoid the fight, so they can't shield everyone easily.
I chose The Black Cleaver over Last Whisper because it uses Zed's massive AOE team presence effectively. When you do your combo in a team fight, you are shredding 10% to 20% of most of their team's armor, and you are shredding 40% of your primary target's armor. The combo is R E Q [basic attack]. At this point, your target (AD carry or AP carry) has 60% of their armor and half their health, so you can finish them off with a basic attack and another E, then you just wait for your Death Mark to finish him off. But the best part is that your team, no matter which angle they are entering the fight from, can use that 10% to 20% armor reduction on the whole enemy team. This greatly helps out the AD carry for his initial burst damage. Also, and I'm not sure about this, but whenever Zed and his shadow attack the same target, I think it applies The Black Cleaver's unique passive for each source of damage, meaning it double stacks. I haven't noticed if this is true for Shadow Slash, but this is definitely true for Razor Shuriken. Reading their descriptions, it says that overlapping Shadow Slashes do not deal additional damage, however Riot does not clarify whether the shadow counts as a second damage source when applying the The Black Cleaver's passive.
Despite my choice, Last Whisper is still a strong pick. Use it, if necessary.
I don't choose Ionian Boots of Lucidity, because you already get decent cooldown reduction from the item build. Instead of focusing on spamming your abilities, you can increase your tankiness, so that you survive the enemy team's mages. I tried this build out against a full AP team that had a Shaco jungler, who was the only source of attack damage. Because I was up against a bunch of mages, I went ahead and got Mercury's Treads. It not only helps out with Crowd Control reduction: it makes you tankier. However, if the enemies' AD bruisers and the AD carry are the main annoyance, then you should build Ninja Tabi.
The Bloodthirster is a critical item. It gives you a solid +100 AD, when it is fully farmed up from minions. This item, when maxed, also gives you 20% life-steal, which is great sustain, farming potential, and it gives you an edge in team fights. Although Zed is a massive AOE caster with overlapping damage zones, this item does not work with on-cast abilities. It only life-steals on basic attacks.
I make it a point to avoid buying Attack Speed items. When you're in a fight, you don't want to worry about autoattacking. Your skills should do the job for you. Autoattacking is an ADC's job. You are melee, and you are not a true bruiser, so you need to use effective decision making, to know when to enter the fight as a burst caster, and not rely on sustained autoattacks.
Blade of the Ruined King is a bad item on Zed, because it relies on basic attacks to be effective. Leave it for the AD carry to buy. You, personally, need a bunch of Attack Damage for your spells.
Ruby Sightstone is a great item to have, if you're serious about getting out of ELO hell, or if you're stuck with a bunch of noobs. A great example is a support who tells you things like "Dude we don't need wards. They don't have a jungler." Yeah? Well what about controlling the lane? You just wanna sit under your tower all day long and get poked to death from the bush? Having a support who wards properly often makes the difference between a good game and a bad game. When your team is losing, wards are essential for getting back in the game. You can use them to:
1. Set traps for enemies wandering in the jungle or river.
2. Defend and track enemies in key locations on the map. If your team needs to farm up before the next fight, you need to ward the Wraith camp's bush, or maybe the two mid-lane river bushes, and give your team a way to safely farm the jungle.
If you have a good team who wards properly, then you don't need Ruby Sightstone. You can safely replace this item with Mercurial Scimitar or another The Bloodthirster, or go ahead and get Last Whisper so you can shred tanks even better. Another good option here would be to get Ravenous Hydra. It gives you good damage stats, and you can use your basic attacks to further increase your AOE damage in a team fight. This item might even proc the The Black Cleaver stacks on the enemies that are indirectly hit by you. This item synergizes extremely well with Black Cleaver and your AOE damage. It will let you shred an entire team solo. But you still have to keep the enemies distracted, so they don't CC (Crowd Control) you.
I pick up an early Q Razor Shuriken so I can harass the enemies and get last-hits on minions. It has a relatively high cooldown, decent range, and it should be used to harass the enemy as well as get last-hits on minions in the back of the minion wave. You can use it to last hit several minions at once, because it has a wide area of effect.
I fill out my skill set by level 3, taking one point in each skill. Then I upgrade my Q one more time, for the extra harass.
You should get a point in Living Shadow early, because it is a good escape mechanism. You need to get it at level 2 or even level 1, depending on the circumstances. If the enemy jungler is Jax he can jump on you and stun you, right after he does the Blue buff. At this point he will be level 2, and you might still be level 1. If you don't have Living Shadow yet, you will be forced to use Flash.
I also get a second point in W Living Shadow, because the upgrade is very rewarding, early on. You increase your bonus attack damage by 100%, versus waiting to get the second point in W until the end of the game. The passive on W increases bonus AD from 5% to 10%, which is a 100% increase. The rest of the upgrades increase your damage for a lesser difference in percentage, so you experience diminishing returns after the second upgrade.
After this, you max out your E Shadow Slash, which is your main source of AOE damage in a team fight, and while farming and pushing lanes. Your E is spammable, with a 3 second cooldown, so this makes it the obvious choice to max, first.