I'm a 2.2k ELO player on EU-W, I primarily play ad carry but whenever I play mid I almost always play fizz, he's one of the most fun ap champions for me and can snowball a solo queue game really hard.

Fizz is an amazing champion for solo queue that can snowball really hard, I like to think of him as the riven of midlane.

Now first, before we talk about how to play him or what items to build, I want to go into who to pick him against, and who not to.

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The main 3 champions not to pick fizz against would have to be Galio, Morgana or basically any bruiser / tank.

The reasons for not picking fizz against these champions is simply that they are too tanky for you to burst down, consider Galio's passive and also Morgana's shield. You also shouldn't play Fizz against bruisers for much the same reason. It can still work but you have to completely outplay them and your short window for kills is in the very early game.

3 champions you would want to pick fizz against are Twisted Fate, Karthus and Ziggs

A common trait of these champions is that they're all squishy, they will all have trouble building enough damage to be useful and enough magic resist to not instantly die to your amazing burst.

Fizz is one of the biggest counters to TF because of his kill potential, TF has no natural escapes and relies on flash to be able to dodge the shark, Fizz's E is also exceptional at dodging stun cards.
The entire point of picking fizz against TF is to constantly keep him in fear of death and stop him from roaming with his ult.

The longer you are left in lane against him the higher chance you have of getting a kill, if you also have the area around mid warded you can successfully stop TF's ult with your own if you shoot it fast enough.

Concerning Karthus and Ziggs, much like TF they have no natural escapes or any way to stop your ult from latching onto them apart from flash. You can also use E when Karthus ults to completely stop the damage, Fizz's E is also extremely useful to jump over the minefield that Ziggs lays on the floor.

For summoners I always go flash ignite, some people like the idea of going heal ignite on Fizz but I don't think the advantage you gain in lane from going heal ignite outweighs the disadvantage you get in teamfights.

Regarding masteries I just use a standard AP page, 21-0-9

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For runes I like to take the armor yellows to minimize the harras you take from the early-game auto attacks. The rest is a pretty standard AP page.

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First back should be 1 or 2 dorans, a ward and some pots.

The first item you want to rush for should be sheen, it greatly increases your damage and also procs off your Q.

Boots wise I can't see myself going any other boots than sorcerers.

I personally like rushing lich bane, although you have no extra AP to compliment it, I feel you still do enough damage and the movespeed can help with roaming.

Alternatively you can get a sheen and build a deathcap straight after.

Once your lich bane is complete you have a choice of two item routes, rabbadons deathcap or zhonas hourglass, zhonas is such a good item on fizz as it synergizes so well with Fizz's E, much for the same reason people build it on vlad because you can zone people for so long with it.
After that you complete the other item you didn't chooose.

I think the choice between Zhonas and Rabbadons depends on the enemy teams setup, if they have alot of stuns or cc to lock you down you might want to take the zhonas, if not rabbadons generally does more damage.

Items to consider can be, guardian angel, void staff, deathfire grasp and abyssal scepter,

So one thing you're going to have to come to terms with and you might not like is that in the first few levels you're going to have a hard time, like any melee champion played in mid it can be difficult to get the early cs when you're being constantly auto attacked and harrased with spells.
You're just going to have to deal with it and let the wave push into your tower if need-be.

It's possible to get kills from say level 4 onwards, but the main time you should be going for kills is level 6 whenever you have your ult. You'll also find the ult makes it extremely easy to get ganks off on your opponent.

Throughout the laning phase you have 2 things to focus on, either roaming or trying to kill your lane opponent, Fizz deals a great amount of damage early game and it can be very easy to roam using your E to hop over walls and to escape from players.

Your main role in teamfights is to be an assassin, depending on who their ad and ap carry is you ideally should land your ult onto one of them. Later on into the game when you have finished some of your main items you'll begin to notice if you got fed enough it can be quite easy to one-shot their ad carry, make sure you stay on the sidelines until you see a good oppertunity to do so.