Welcome player,
Obviously you are here because you want to learn to play Katarina.
This is by far the most effective build I have found for her.

Mark: Hands down go with Magic Pen

Seal: 4x Dodge runes; Flat HP or scaling. Your preference. I go with scaling.

Glyph: Go with AP here. Scaling or flat again is your preference. I go with scaling.

Quintessence: Flat HP bonus. For the early game, as you will be gaining lots more HP with the items you make.

Katarina's abilities are very potent so you want to go down the defense tree.
Your entire goal is to gain stacks and not to die!

To start, get a little AP so you can last hit creeps more effectively as well as to build into your spell vamp. Around 800~ gold, you want to head back and get your hextech (You can wait for another 350 for boots, but by about this time you'll be low on health and need to head back anyways). This is very important, because you do MASSIVE magic damage and will regain plenty of health with your abilities. After, the items speak for themselves. You'll want to get Soul Stealer so you can start ganking and start really gaining some AP.

(Tip: You'll want your upgraded boots quickly after SS, as they give you the Magic Pen you desperately need. If you have enough, get one of the following health items so you have more HP and won't die as easily. Katarina is easily one of the most targeted heroes in the game. So you want to make sure you stay alive!)

Early Game: Katarina is horrible during the first few levels. With barely any damage output, you want to stay back gaining all the XP you can and last hitting with your "Q" to the further away minions and attempt to melee and "E" the closer ones

(TIP: Your Shunpo always leaves you on the opposite side of your target, so running to the side is a good way of not getting into a bad spot, as it leaves you on the other side of the minion and not behind it. IF you can get behind the minion safely and "E" hit it you'll end up in front of it, which is your best move.)


Early Mid Game: By now you should have quite a few levels in "Q". This is great because it's an awesome harassment/farming/pushing spell. You'll really want to start trying to hit the enemy hero(es) with your Q (even better if you have W active) and keep them down. Once they get low enough Shunpo in and let those daggers fly with "R".
(TIP:You see the enemy hero with their pot up? W+Q is an awesome way to null that out. Doing most of the damage that pot is going to heal + making the health pot only 85% effective overall.)

Mid-late game:(Roaming phase) Now's your time to shine. Remember that you are a high priority target to the enemy team, so you always want to be one of the last people to come in to team fights. Your "R" does hugeeee damage but only if you're hitting people with it. Meaning you need to stay back and let those stuns, silences, and knock ups be used on other team members so you can really do some damage. Now you can be in the midst of a team fight too just don't use your death lotus until till you can for sure not be stopped.
(TIP: Once a team fight starts, you want to make sure that you get assists on the minimum for your SS stacks. So make sure you throw your Q into the majority of the enemy team so that you get those crucial stacks! Plus if activated with W, you make all healing effects 50% which is ****ing awesome.)

I am publishing this guide to get a little feed back on my build. It's not entirely done as it requires a lot of typing but the general idea I think I have gotten across so far. Thanks for reading and happy hunting!