Hi,
I'm Maurice. I'm playing since Season 2. I`m maining Kata since Season 5 and got ~2000 ranked games with her. In Season 6 she carried me to diamond. Since her rework I'm spamming her everyday so I can say: With my experience with the old Katarina and many games I played with the new one so far I'm already pretty good with her. Right now when im writing this guide I'm ranked as #245 Katarina in the world on LoLSkill.net.
Even if this guide isn't the longest it still contains all the important information you need to know. I'm sorry if my english sometimes makes you wanna punch a wall. I'm not a native english speaker. If you got any suggestions how to make this Guide better -feel free to comment. It's my first Guide and I'm happy about any feedback.

+Extremely fun to play
+Good for being a OTP (Steep learning curve)
+If resets are managed correctly you feel like playing beyblade
+Resourceless
+AOE Carry

-Relatively hard to play now
-Still hard countered by CC
-Weak vs Tanks

Most of the masteries are selfexplaning. The only masteries you could replace are...

... Fresh Blood with Feast - I like the stronger AA much more because with your E reset you're able to jump in every few seconds to trigger Fresh Blood .

... Battle Trance with Bounty Hunter - I personally prefer Bounty Hunter when the enemy team has many squishys.

... Savagery with Wanderer - If you feel like you don't need the help with farming pick Wanderer .

... Assassin with Secret Stash - Completely viable, though I would take Secret Stash in a poke matchup.

The runes shown above are for an AP matchup only. In an AD matchup I would pick:

You may replace the MS Quints with AP but I prefer additional dodge, roam, chase and disengage power.

For all those who aren't using smartcast ... you should use it! Especially with this Champion. You want to use your resets as fast as possible and that is easier with smartcast.

(Passive):( Voracity is Katarina`s passive. It reduces all your spells' CD by 15 seconds whenever a champion you damaged in the last 3 seconds dies which means you get Q W and E back instantly. This passive is the reason for Kata players being called out for smashing their head on the keyboard. With the new Kata you have to manage your resets properly or you will go down while looking like an absolute idiot.
Always remember that you got a way out when you make a kill. Fancy flash EQs under the turret and E back onto a minion can make you not getting hit by one turret shot.
If you got a few resets in a teamfight and you got all your spells out between the kills you basically feel like beyblade.

Her second passive is the DAGGER damage that scales with AP, AD and level.

(Q): Bouncing Blades is your main poke and preparation tool. It bounces on 2 other targets after the first one and places a DAGGER 350 units behind the first target in the angle you hit it. Use it for poking but be aware that you push your lane when the champion you poke is standing near minions. Notice that picking up the DAGGER deals more damage than your Q. Sometimes it can be more effective to use Q on a minions so the DAGGER lands in the way where the enemy is moving.

(W): Preparation throws up a dagger that lands after a short duration and gives you a great short movement speed burst. This spell helps to chase, disengage, jump in and have a way back out and to prepare a surprising turn-around by placing in while disengaging to jump back and burst the enemy down.

(E): Shunpo is the best part about Kata. It's an instant jump/dash/TP to a targeted enemy, allied or Dagger. You can choose where to land around the target which helps alot with picking up DAGGERS while still being in range for the DAGGER to deal damage to the enemy. Picking up DAGGERS reduces E's CD extremely but not to 0!!!! It is also an AA reset which helps especially in the laning phase where 2 additional AA's can win you the lane.

(R): Death Lotus is and AOE-DOT that, if not interrupted, deals an insane amount of damage. Only use it when the enemy has no hard cc left. Notice that the use of summoner spells that don't dislocate your charakter (ignite,barrier etc.) wont´'t cancel your ult. On the other side all item actives cancel your ult (don't use gunblade while spinning).

Tipps:
#1 You can jump to the spot where a dagger will land before it actually landed.
#2 Your E is an AA reset
#3 Your Dagger lands behind the target you hit first
#4 When you use W and use E in like 0.1 seconds before it lands you deal the DAGGER damage on around your E target

- Walljump:
W near wall - (If you have enough time wait for the dagger to land so you get the reset) E on the wall on the outer border of the dagger

- Safe combo:
Q - W - E on Dagger but near enemy champ - trigger Dagger - E on W

- All-in Combo without Ult
Q - E on Dagger but near enemy Champ - W - AA - E - AA - E - AA

- All-in Combo with Ult
Q - E on Dagger but near enemy Champ - W - Ult

Most is explained in the matchup chapter. Before you engage on the enemy make sure how you want to fight. If you think he will outtrade you if you're near him for too long place a W before so you can jump back.
Be aware of ganks and of the map. You need to stay safer now because you're missing your amazing wardjump you had with the old Katarina. Sometimes a jungler comes with half HP and your midlaner is half HP aswell. Try to place a W and you may be able to turn around for a double kill. If you think you can kill one of them you should engage.

Don't engage a fight except when the enemy team has no instant hard cc left. Do the "All-in with Ult" Combo when you're behind a wall so you can surprise the enemy team or when no instant hard cc is left.
When your Team is getting chased place your W and when the enemy team is clumped together do a surprise E-Ult.
When you see an opportunity for a reset go for it but don't get too greedy. You should always be calm and not trying to be the hero we don't deserve but we need. Ah! Screw that! Have fun and jump around!

Roaming is essential for Katarina.
When you pushed out your lane and your enemy midlaner is not in sight then it's your time to shine. Even when you're getting spotted by a ward you're still giving your laner pressure. The enemy will respect the possibility of a roam. Remember that your Dagger lands in the angle your're facing your target so it sometimes can be better to Q a minion so the Dagger lands in the way where the enemy is most likely to run. Your passive deals much more damage than your Q.

As soon as I am able to produce good ingame footage of some combos I will add them in this section!