Fizz is a tricky melee assassion and fighter, full of redirection, distraction and speed.He can sneak into battle and burst down any squishy target or if correctly build he could stand toe to toe with the tankiest champions.
Abilities
I suggest startingwith playful trickster first.This will give you the ability to escape an invade or an early gank.Then max Seastone Trident as it gives u huge burst damage and power early game laning.Urchin strike is your second spell to max after Seastone Trident.It is great for engage and could sometimes help you avoid deaths if used wisely.Ofcourse learn you ultimate ability at level 6 ,11 ,16.It is great for engage or when you have to chase your enemys as it slows down anyone whos atyached with it.
Laning
Fizz's early laning is extremely powerful if you are willing to take some risks.Poke your enemys with Urchin Strike and Seastone Trickster and chase them down with Playful Trickster if you can get the kill.Fizz's passive will help him survive longer as he takes reduces damage from basic atacks.Even though many players do I do not recomment using Playful trickster as a engage weapon because it has a big cooldown and when it is unavaible Fizz could be easily ganked and killed.Keeping your E will make sure that u cant be killed in early game by jungle ganks which is very important.Fizz is extremely powerful againts squishy range AP carrys such as Brand, Anivia, Cassiopea, Twisted Fate and many others.However he has his weaknesses.As a melee midlane champion is it very likely that you will have less minions farm then you opponents.This isnt something to worry about as FIzz is very strong in fights so you should be able to get some kills in early game.
Runes, Masteries and Spells
For Runes I take magic penetration marks flat ability power seals magic resistance glyphs and flat ability power quintessences.This gives me a lot of MR to survive and be able to poke on midlane and also gives me a lot of ability power and good magic penetration to be able to score more kills.For masteries i take 21/9/0 putting 21 in ability power and 9 in armor and mr aiming to be as durable as possible in early game.For spells I use ignite and barrier.The reason i take barrier is that I do not need Flash when I have Playful Trickster and Barrier helps me score at least 1-2 kills more per game as it enables me to turret dive way more easily.
Build
I start with Boots and 3Health Potions.Then I build a Cathalyst the Protector which gives me good potion of mana and health that I transform into Rod of Ages right after that.Then I go for some mroe ability power.Rabadon's Deathcap gives me a lot of damage and then I build Lichbane.At this point FIzz has enough damage to kill anyone in the game and is able to kill squishy targets such as adc's in just one Q, W and a basic atack boosted with Lichbane.Therefore at this point I recommend building some durability such as Zhonya's Hourglass and Abyssal Scepter or a Warmong Armor depending on my enemy team.