Fizz is a powerfull mid lane champion because of his signature double skill "Playfull/Trickster", that skill gives him the ability to become invunerable and untargetable to any Skill, atack or Turret. It also gives him the ability to jump by some parts of the jungle with the "Trickster" and gives him high mobility to gank and to flee ganks in the mid lane.
Fizz "Urchin strike" deals a massive damage to the enemy by the same time it's a effective get closer, this way gving Fizz a high advantage in the fight as he is a melle champio.
Fizz "Seastone Trident" is the best skill in the lane phase, and gives high damage later in the match. The passive ability of this skill, grantz Fizz a Bigger damage on minions and champions per auto-atack, this way facilitating farming in the lanes and increasing the damage Fizz deals in a 1 on 1 battle.
Fizz ultimate ability "Chum the waters" Puts the enemy in a area slow for a short amount of time and then grants a knock up with high damage to all enemies caught inside the skill area, this skill is Fizz combo starter in my gameplay, because it has low CD and gives you a big advantage uin the start of the fight.
Fizz Passive ability "Nimble Fighter" is very underated, it is actually a very important part of Fizz gameplay, With no colision with minions Fizz is able to farm and dodge from skills much easier than other champions, not only that, Fizz can decide to atack the enemy at any time he wants because of it.

To deal high damage to an enemy as Fizz you must do your combo the right way, and always have in mind that in the early game till you reach lvl 6, you will most probably not kill a enemy in a combo, so It's not good to dive at this time.
lvl 1, you will only have your "W" ability to use, so unless the enemy gets close o you, you don't atack, and only activate your "W" to fight.
lvl 2, You get your "Q" ability and becomes possible to deal high damage to your enemy, for a example in level 2 against full squish mid laners I am ale to take away more than 50% of the enemy total HP in only one "W"+"Q".
lvl 3-5, You can use the "W"+"Q" and then use "E" to flee without takning damage if you think you're not gonna kill, or use "Playfull" to deal damage and slow your enemy if you think you can kill, but beware that Fizz is easly killed once you waste your "E" ability, so decide when is the right time to use it based on your enemy skills and situation and in the last apearance of the enemy jungler and yours.
lvl 6, With the "R" ability now usable you can try two diferent combos to kill your enemy, and if you decide to atack, go on with the atack till the end. The two combos are:
N°1 - Use your "R" when the oponent takes a point to farm a minion or to cast a spell, at the time "R" hits The Target, You use the "W"+"Q" combo and hit him twice with normal atacks, then use "E" "Playfull" to damagehim and slow, If needed Your "W"+"Q" will be ready to be used again and that should be enough to deal with most of the AP mid laners.
N°2 - Use your "W"+"Q" combo to get closer to the enemy and right after it use your "E" "Playfull" to deal damage and slow, Retreat a little while on slow and cast "R" after that hit him and deal the "W"+"Q" combo again.

All I'm gonna do here is explain the importance of the items I listed to be done in the early game as Fizz, many people tell me that some of the items I do are really not needed and that I should change that, but in the early game, the little status modifications are important, thats why we buy runes and should be why we buy early game items too.
At the start of the game you need to try to go back only when you have money to buy the following items:
-Double Doran Ring
-Null-Magic mantle
There is the Item many peolple tell me not to do, the Null-Magic. But I have my motives to use it, in the past seasons Null-Magic was part of Fizz core Item Lich Bane, and I used to buy it to make the damage dealt on Fizz te less possible in the mid lane, people say that this don't change a thing, but it's actualy not true, how many times have you ended a fight with less than 100 HP because of one hit ot one skill? every single life point you save is important.
After that I normally buy a Kage lucky Pick to be able to start rushing my Items. May people say that doing this as late as I do is useless, but it really is not, with that build I have a strong early game and icrease my speed to get to the late game with the extra money.

Pros:
High mobility
High Damage patterns
Invulnerabillity

Cons:
Close combat
Squishy
Weak on early game
Team fight target.

Fizz is strong against AP casters, but the ones he's best against are:
Karthus
Morgana
Lux
Veigar

Fizz is weak against AD mid laners and Stun/Silence casters such as:
LeBlanc
Talon
Teemo