I've been playing Katarina a while in solo queue, and it really hasn't gotten me anywhere due to feeders, quitters, etc. But ever since I found actual team mates and formed a team. I've been able to fully feel,coordinate, and carry my team mates using her. So I heavily recommend finding if not good, at least dependable team mates that you trust and have synergy with.

Now, here are the main elements in using a Katarina;

Okay, the first thing you need to know about Katarina is her AP Scaling. Because I can say frankly, that it's broken. So broken that her ultimate scales nearly twice your AP.

So let's say you have 800 AP.
Her maximum damage for her ultimate would be 700.
So 700 + 800 x 2 = 2100, with just her ult.

Next would be her passive. Her passive resets all of her cooldowns by 15 seconds everytime she gets a kill or an assist.

Now keeping that in mind, the third and most crucial thing about Katarina is how spammable her combo is.

I recommend either: Q -> E -> W
Or: E-> Q -> W

And of course once her ultimate is up, just add it to her combo.

First start off with an Amplifying Tome and a Health Pot.

At level 1, get your Q first. Remember that your Q hits up to four nearby targets. So keep in mind that you don't have to aim directly at your enemy to harass them. But instead use it on a nearby enemy minion to racket up creep score as well. Two birds with one throwing knife right?

Now, the moment you hit level 3 (yes even at level 3) your kill potential increases massively, provided you can land your combo properly and easily, you'll be able to do around 200-300 damage.

Also keep in mind that you do not need to always need your E. Being too dependent on it for reacting may cost you. Remember that your Q give you the bonus damage needed once you land your W, and that W gives a small movement speed needed to gain distance after harassing.

Keep using your combo properly and you'll hit first blood for sure. Even though you've been harassed hard, just be sure to be able to exchange blows as well, and you'll be able to have the upper hand due to your higher damage output than most enemy midlaners.

Once you get your R, you'll just need to harass the enemy till he gets up to 40-60% health. Then spam your combo to collect the kill.

Go back to base and build your Mejai's Soulstealer, Boots of Mobility (Or just boots if you don't have enough money) then grab as many Health Pots as you can.

Now: Grab a kill onto the mid lane and collect your first stack, then proceed to gank either the top or bottom lane in order to both help the lane, and farm stacks.

Simple, either have a jungler gank your lane, or if you don't have a jungler, gank an enemy lane to collect stacks and regain exp and gold as well. However, also be aware of counter ganks from both the midlaner and enemy jungle, and make sure to keep and eye on that minimap as well.

Remember, that coordination with your teammates is key to ganking and killing their lane. Failing the gank will cause your gap to be even bigger and may even bring down the lane you ganked.

Cont.

Once you've killed either lanes, go back to mid and gap your enemy mid laner as hard as you can by killing him again or forcing him to go back. (be sure to be able to hold your lane as well.)

Go back to base, and buy a Giant's Belt to transition later into Rylai for the slow and HP, Health pots, and then homeguard to be able get back to lane and farm more stacks.

Just rinse and repeat until the Laning phase ends. By this point, you should have 8-14 stacks and should be able to build your Rylai.

Just continue on to build a Void Staff to shred their Magic Resist, a Deathcap and finish off with a Zonia's Hourglass to prevent getting shutdown.

Now, during team fights. 3 things are crucial and WILL decide if your team will win the team fight or not.

1.) Timing- Wait for the enemy to engage and use their cc that can cancel your abilities, but don't wait out too long as some members of the enemy team may already have reset their cooldowns.

2.) Your Placement- Your positioning when using your combo-> ultimate, depicts how many members from the enemy team will be hit and if your cooldowns will reset. Again, Katarina depends on her resets as it gives her the damage output she needs for her combo as well as her ultimate.

3.) Coordination with your team-
This may be the most important.
For this will depict and control the previous two I just mentioned. This is why, I personally NEVER solo queue, and instead, wait for my friends to come online so we can have a minimum of 4 people which leaves only one liability regarding coordination.

Now, do all of these correctly. And you should score a triple- pentakill.

By this time, you should have at least 800 AP (Provided you have complete Mejai Stacks)

And if you still can't get a penta..well. That's just your teammates taking you kills now.(If you're in a team.)

For the past 8 games I've played with my team, I've only lost twice, and my deaths have not gone up 4 and my kills have remained constantly above 14. And getting 6-10 kills by 10 minutes.

Likes I said, it all comes down to team coordination. So to everyone reading this, I hope you find a good team out there!

Good Luck in the Fields of Justice and Thanks for Reading!
-BRS. Beast v.2 (Clickophobia)