Fizz is a melee assassin that focusses on oneshotting squishy, but also tankier champs. His favourite meal are ADCs, weak midlaners and squishy supports (except Nami she is your fish friend).
Another trait is that he is slippery and can easily go in and out of a teamfight.
He is what I call a self-sufficient champ because he can solo engage, solo kill as well as solo farm and escape. You don't have to rely on your team as hard as with other champs, like Aniva for example.
Outplays are common and easiest to do with Fizz Playful / Trickster

With Fizz, you just want to all in engage in teamfights, and if you did well early game, you can easily oneshot every squishies, even more robust champions like Graves.
There are three different behaviours with Fizz:

• Trading
• All-In
• Teamfighting

Trading
You keep repeating your harass to make your burn more effective, also you can oneshot easier then. Just go in with Urchin Strike, activate Seastone Trident before and then use Playful / Trickster either to go out or to do extra damage, followed by auto attacks.
This way, Fizz can even trade under the enemies turret.
All-In
Just activate everything you have, Chum the Waters first. You have 20% more damage, Q and E. If that isn't enough, the enemy will have a strong bleed because he is low and you have Seastone Trident's burn. If that isn't enough either (then you did something wrong), you still have Ignite.
Usually, it is enough to do a WQ combo 2 times, and then you can go all in either Lv3 or Lv6 and get the kill. Your all in kills every Midlaner below 2/3 of their health.
Teamfighting
This if Fizz weak spot. You have to be fed to oneshot a single target in a group, and most time everyone is still full life before teamfights. If you can oneshot someone or do enough damage for your team, try to hit Chum the Waters on their backline. Then go all in (that's the only thing you can do in teamfights, you can't harass because you are melee).
The good thing is that Fizz is slippery. The best thing that can happen besides oneshotting someone is to do huge AoE damage with Chum the Waters and Playful / Trickster. Then, you can use Zhonya's Hourglass to wait for your next E or just dash out of them with Urchin Strike.

When to pick Fizz

Pick Fizz whenever the enemy midlaner picks a normal mage, such as Lux, Brand, Orianna or Ziggs. Also, Fizz is stronger than most Assassins like Talon, Katarina or Zed, but you have to know how to play against these champions and the matchup is harder. Irrespective of their midlaner, you can also pick him into squishy supports like Soraka or Sona, immobile Adcs ( Jhin, Ashe, Miss Fortune) and Teemo Top. Just make sure to roam your victims enough.
Here is a list of champs that I would pick Fizz into (they should have at least 1 of them and the rest shouldn't consist of hard matchups):

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When NOT to pick Fizz

Don't pick Fizz into tanky midlaners such as Swain, Cho'Gath, Pantheon or into duelists like Diana. You should at least play defensive when the enemy team has much map pressure from other lanes, such as Shaco, Pantheon, Tahm Kench. Avoid picking Fizz into hard harass champs like Diana and Viktor. You can stomp these champs, too, but you really have to know how. Don't pick him into full tank comps, e.g if top, jng and supp is tanky.
Here is a list of champs that I would NOT pick Fizz into (if they have at least 1 of them and/ or the rest consists of hard matchups):

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Pros & Cons

SPACESPACE

Pros Most powerful all in in the game One of the most mobile champions overall Easy to pick up Makes outplays easy, can dodge every skillshot Slippery Can absolutely dominate enemys Much utility in items and skills Self sufficient champ for SoloQ Many people can't ouplay him One of the strongest roaming champs

SPACESPACE

Cons Horribly weak pre Lv3 Hard in teamfights Hard to master E timing is essential and can backfire Dead if cced Hard to come back when behind You have to know what you are doing when Playing aggressive is risky and can backfire Fizz can be outplayed hard, negating all his dmg You have to learn his laning and roaming behavior

To start at lane, I would always recommend Corrupting Potion. Fizz weakness is his Lv1 and Lv2. He can't trade there, because he takes more harass than he does plus minion aggro. Also, the enemy won't let you farm because you are melee and take all the harass. Try to lasthit as well as you can and only take safe farm. When you take some damage, it's ok but you shouldn't use more than one potion pre Lv3.
Depending on the enemy laner, you will fall back in cs or fall hard back in cs. It is even better to stay under tower pre 3, only having 5 cs then taking too much harass and going back.
Against champs like Lux or Brand, you can skill your Playful / Trickster first to dodge their skillshots and farm a bit.
When things are going well, you can go back at 900 gold to buy Lost Chapter for the mana regen and ap. If not, go back at 350 gold to buy a Dark Seal which extends the duration of corrupting potion. If you have some gold left, buy Vision Wards.

(Early game is pre Level 3 and after your first back)
Instantly after Lv3, you can start playing aggressive. You should be nearly full life and still have some Corrupting Potions to clearly win the trades.
But bear in mind - playing aggressive is always a risk. Especially when the enemy is ahead in cs and has more items.
Also, you are vurnerable to ganks. Place Stealth Wards at around min 3 and Vision Ward in the bush behind the blue buff (at the river) to prevent ganks.
When trading, Fizz is more persistent because the longer trades go and the more often you trade, the more the enemy will bleed through Seastone Trident. I recommend playing Fizz as often as possible to get a feeling for trades and what you have to do.
For short trades, go in with Playful / Trickster do damage the enemy and weaken the minion wave, then use Urchin Strike on a minion behind you to dash back.
For longer trades, engage with Urchin Strike and activate Seastone Trident before. Then use Playful / Trickster, followed by autoattacks. This way, you normally win the trade, but you are left without defense. As I said, you have to get a feeling on when to do this and when not. This trade is like a double-edged sword because it creates insane pressure but makes you killable. When this trade succeeds, you can probably all in the next time and get a kill. Always use this trade when you feel safe and you know their jungler can't gank you.

When the longer trade made the enemy below 2/3 of their hp, you can repeat this tade. When using Playful / Trickster, Ignite the enemy straight away to amplify the burn from Seastone Trident. When the enemy flashes, Flash, too. With a few autoattacks, you get the kill. This can work at Lv3 already, but is most likely going to kill at Lv6 with your Chum the Waters.

(Mid game is post Level 6 and after having reached at least one core item)
This is where you can start roaming and oneshotting. When your k/d is positvie and you have Hextech Revolver / finished Morellonomicon / Lich Bane, you should roam bot a lot. This means that you have won your mid lane and the enemy midlaner won't be able to solo carry, and when u roam bot, you can stop the enemy adc from carrying, too, meaning that you nearly took the entire carrying and winning potential from the enemy team.
When roaming, this oneshot combo is usual:
Engage with Chum the Waters, followed by Playful / Trickster as gap closer. Urchin Strike with Seastone Trident should be able to finish the adc.
If you have flash up, you can also use your Playful / Trickster and flash on the adc before you land. Finish with QW.
Get as many kills in the mid game as possible, because there your kill potential is at its highest. When you are really fed, go full ap ( Rabadon's Deathcap, Void Staff, Luden's Tempest). If the enemy team still has damage output, go Guardian Angel and Zhonya's Hourglass.
Pick up the midlane farm as well as possible, and when you see their midlaner pushing your lane, all in him.
When roaming and rotating and you have vision over their jungler or support roaming, you can also camp in bushes to throw Chum the Waters out and all in them. Because they don't see you, they get to close so they can't dodge your R and E.
This is why I don't upgrade my ward into anything. The Farsight Alteration is useful for squishy mages that can be oneshotted easily. But when u ward a bush and enemies camp you, u can escape easily in contrast to mages. Plus, Farsight Alteration wards cannot be used for tp and can be seen by enemies.
The normal Stealth Ward in contrast provides you with 2 invsible wards that can be targeted for tp and give more vision (which is useful for following enemies and oneshotting them).
After having killed the enemies, group with your team to take objectives (e.g after having killed their supp and/or adc, go take bot towers). While your team is doing drake, herald or baron, camp in bushes to see the enemies comming and oneshot some of them.

(Late game is when both teams are constantly grouping and teamfighting only)
The late game is the hardest phase to learn with fizz. One mistake can easily throw your game due to longer respawn times. Be careful with engaging and don't just blind all in. You get a feeling for this when playing a lot of Fizz games. You can engages yourself, but it's better to wait for your frontline to engage.
Make sure that the enemy's frontline is close to their backline, so Chum the Waters AoE hits them all. You can do further AoE damage with Playful / Trickster, which should nearly kill squishies if you are fed. To avoid getting focussed and oneshotted then, you can either use Urchin Strike to take distance from the teamfight or you can use Zhonya's Hourglass. If you have bought Guardian Angel, you are even harder to shutdown.
After that, you can re-engage and clear up what's left.
Teamwork and communication is important in order to not screw up, becase Fizz can't dive alone into 5.
When you are still behind in late game, this is a problem. Hope for good engages from your team and mispositionings in the enemy team, and try to pick someone out and all in him. Try to stay grouped and output as much damage as possible, to win teamfights and then take objectives.

There are some things to know about Fizz skills, which can be seen in this video: Play Video

Also, here is a full list of spots Fizz can jump over and spots he cannot pass: Play Video
I have a whole channel only dealing with Fizz-related content, so if you want to be a Fizz nerd, you better check out my channel.

• Know your damage. The damage output is so high that noone expects it.
• Learn when to use E. Timing is essential.
• Play defensive pre3 and aggressive af post3.
• roaming and getting kills > cs