Keep in mind that this guide is not finished yet. (work in progress)
Before we start this journey I need you to familiarize yourself with some of these terms that I will be frequently using throughout the guide. If you are unsure of anything please do leave a comment and I'll get back to you as soon as I can.

AP - Ability Power
AD - Attack Damage
MR - Magic Resist
AS - Attack Speed
CDR - Cool Down Reduction
CD - Cool Down
Gank - Term used when you try to pressure a lane
Counter Gank - This is when you see a lane getting ganked and you attempt to assist with the aim of turning it around.
Match Gank - When you see the enemy jungler in a lane and you know you can't take the fight, you aim to either trade even or protect your laner from dying

Fizz is, if played well, a very strong jungler. Fizz has some CC, and a very high mobility. When ganking, Fizz can easily close great gaps, and deal much damage. Counterganks or counterjunglers are not able to follow Fizz jumping through the jungle.

The damage output with a spellblade passive item is insane, you can 1v1 almost every champion.


Pros
Great damage
Strong vs almost everyone
Nice gank potential
High mobility
Ok sustain

Cons
Squishy
Item-dependent
Hard to catch up when behind on items

You should start the game by asking your teammates to ward pixel brush (1:10/1:20sec into the game) in the river at the opposite side of the map from where you are starting (red). This wards grants you and your mid laner vision/awareness if the enemy jungler tries to do a early gank mid or try to take your blue buff.

Route:
Start at red buff and ask to get leash from closet lane. I recommend not to smite the red, but instead save it for counter jungling.
Try to gank mid straight after red if possible, watch your vision from the pixel brush to maybe get vison on enemy jungler and counter gank if enemy jungler ganks mid and take his blue if he goes for your blue.

Try to get atleast 900 gold before you go back to base, becasue it's really important to get your jungle item as quick as possible, since it will improve your clearing speed significantly, and for the extra gold of course.

Just start attacking monsters with your Seastone Trident and Urchin Strike. Use the Urchin Strike short after Seastone Trident is activated, to get the bonus damage. If you have blue, use Playful / Trickster to finish or start a camp, without blue it is a waste of mana. Remember that you W and Q are both aa reset, try to always use them seperatly straight after a aa. For example: aa+w+aa+q

Pre 6 / without the ultimate ability:
Try to get behind the enemy or from the side. When entering enemy's vision use you'r Q as a gap closer and try to hold on to you'r E for as long as possible to either juke any cc that the enemy may have or to chase/follow enemy if they use an escape such as a flash or a movement ability. If nessesary use ignite to help finish and secure the kill.

After 6 / with the ultimate ability:
This should be done in two different ways.

1: You can 100% hit the enemy with your ult before entering or right after you entered his vision. Then follow up with the "pre 6" gank.

2: You follow the "pre 6" gank but you cast your R mid Q on to the enemy. If done correctly this is a secured hit with your R.

So the first ability I want to discuss is your E ability, probably the most versatile ability in your kit. Your E can be used to jump over some jungle walls, My most used are the walls near Baron/Dragon pit, the wall behind krugs and the wall near the raptor camp. There are many more but with blasting plants you wont need to worry too much about them (Just experiment and have fun).

The second trick is to use your E to avoid damage. Fizz gains brief invulnerability when you are still on your trident, which can be used to actually cancel some abilities or negate tower shots and auto attacks. The best example I can think of is a Twisted Fate Gold Card on his Pick A Card Ability. If timed right you can dodge the stun completely and avoid the damage. The most important thing to remember when doing this is your next move. Yes you can avoid damage but you are also left in a pretty tight spot. You have a very short range on your reactivation of your E, so a well coordinated enemy team can then set up and pick you off. The best thing to do in that situation is to actually to not use the second active on your E, but to rather let yourself fall on the spot. The slow as a result of this can save you or teammates in multiple situations.

The last thing you can do with your E is simple. As you are falling off your trident you can use Flash and change your final landing spot. The best illustration of these tricks is by Westdoor against Faker in Worlds which resulted in Westdoor getting the kill ---> Click here to watch .He uses his E to avoid a turret shot and then flashes before he lands to get the slow on Faker.

A note when using your Q. Since you dash a fix distance when cast you can use it to pass through some jungle walls much like yasuo. Placing a ward over a wall and Q'ing to get over can help you escape or cut off an enemies escape route. Additionally, since the dash distance is fixed, you can safely use the ability right next to an enemy to gain a bit of distance when leaving a fight. What I mean by this is when you are standing right next to an enemy you can use your Q to put distance between you instead of closing a gap. This is generally used when you all-in an enemy, you would R the enemy, E to close a bit of distance, W for the bonus damage and then Q to escape if you were in turret range or to block your opponents retreat. This is highly dependent on your landing spot after you E since it would determine the direction you will Q towards.