Things I have historically found effective against Zed (disclaimer: I've never broken out of silver and mostly play against bots at the moment. I haven't faced him in his current incarnation. I think this advice is still useful, but feel free to disagree):
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Pick long-ranged wave clear and push early. You'll miss some farm, but you'll stay current with xp, which is what really matters against assassins, and it's much easier to dodge Zed's poke when you're not trying to last-hit under tower. Your primary goal here is to keep up in levels and not die level 3. I like Morg, but you do you.
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Level 3 gets him w, and a huge powerspike. When he has his w placed, stop trying to last-hit backrow minions and just keep moving and watching until he uses double q. People talk about how long his w cd is but it's under 30s at every rank, so he gets at least one w every wave. Keep that in mind.
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Levels 4-5 Just keep pushing him in (while standing behind the Lane's half-way point) and stay warded up. If you're getting low on Mana or under 3/4 hp without hp pots, shove as hard as you can and go back. Keep in mind that at these levels you're stronger than him when his w is down, except in a melee-range all-in.
3.5. For your first back, whenever it is, buy a cloth armor and at least 2 pots. I don't care how much gold you have, you absolutely need these. I don't care if you keep running oom, use fewer spells and get cloth armor. Get a
asap (you don't have to finish the zho ya's right away) and strongly.consider getting another cloth armor just to completely negate all his Lethality. I cannot emphasize how important this is.
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At level 6, try to stay out of his ult range - keep your distance from his shadow, and try to force him to w + flash to get ult off on you. If he ults you, and you aren't 100% confident you can outplay with cc or dodging his Q, wait for him to appear again and flash under your tower. If he hasn't forced our your flash with a gank or by you standing too close, it's really easy to deny his first ult all-in by just flashing, and that really slows down his snowball. You should have stopwatch from the rune by the time his ult is up again, which denies his second huge opportunity to get a kill and start snowballing, and by the time his third ult is up landing phase is pretty much over and you should have adc + support in your lane anyway, making it much harder for him to make plays.
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Once you have your seeker's, your champion's core components (
or
or whatever), boots, and maybe a cloth armor, you're in great shape! You probably still shouldn't pick fights with him, unless you're super confident, but you should be relatively safe shoving from a distance.
"But Zed doesn't even use ults on me, he just roams bot and gets a free double-kill!"
Keep vision up as best you can and keep pushing, you'll KNOW when he roams because you're keeping minions against his tower, and you'll either pin the enemy jungler mid to stop your push (in which case ping up a storm and your bot should be able to fall back easily enough, maybe set a trap with YOUR jungler) or you'll get some serious tower damage off, and your jungler can maybe help you rush down first turret - unless he gets a ton of kills, the roam will actually set Zed back in Co and maybe even gold, depending on how many waves crash into the tower, and you should be able to (cautiously) follow him if he roams again.
So, the highlights:
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Three times to really be wary of Zed are level 3 and first two ults. Level 3 kill can be denied by just sitting back, first ult by sitting back and if he ults flashing the second he reappears, and second ult by stopwatch. Beyond that, general cautious play and aggressive (but long-ranged) pushing should keep you safe.
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If he roams, SHOVE the wave under tower. Don't hit the tower until you know where he is, and be aware of where the enemy jungler might be (e.g. if it's Nocturne with ult up don't sit on the tower if he can solo-kill you) but try to deny Zed farm and just ping like 5 times straight if you think he's roaming.
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1st back get cloth armor + AT LEAST 2 pots. I like 4 tbh. Get seeker's asap and consider another cloth armor, it makes him so incredibly less scary and you can always build ninja tabi or some shit later or even sell it.