Fizz deals extreme amounts of damage. He has great mobility, catches up and kills quite easily, even in the jungle.

Initiate with your shark whenever you can. It slows enemies, hits a good amount of damage and stuns them for a moment. Then use your Urshin strike to hit them and get your DoT on them as soon as possible. Use your W to get a good hit and keep playful/trickster to finish them off when they try to run or to deal damage if they are staying. [Here's a Trick : if they start running but are still in range of playful, just click where you want to land and Fizz will get off his trident with the big AoE towards where you asked him to. If you use the skill a second time you do not benefit of the slow and you get a smaller Area of Effect...]. After you get some cooldown do not hesitate to initiate with playful/trickster. Remember you are quite squishy though, so try not to waddle around alone and if for any reason you get in trouble you have the tools to run. In late game just turn around and unpop the opposite champ if he is alone, try to get some kills before dying if they are more than 2.
Have fun !