Who
I have very close to 300k mastery on Katarina and have played a fair amount using this build, I'd almost daresay equal or more games of Tankarina than AP Katarina. I got Mastery 7 using this build too, it doesn't go too bad.Why tank Kat?
On AP Katarina, generally you jump in and dish out a tonne of damage in a really short period of time. While that's fun, I love using Katarina to dance around opponents and outplay them. Tank Katarina let's you do just that, without looking like an idiot and getting 1 shot as punishment for that. Early to midgame she can be a super mobile, fairly tanky inescapable fighter due to her low cool down dashes and abilities. Seriously, I love playing this build.

Trinity Force
This is your bread and butter item, and is infact the entire premise of this build. It grants you a huge variety of stats. The 20% cooldown reduction is a huge boon. It also has it's passive, where after casting a spell, your next basic attack deals bonus damage. Katarina has all the tools in the world to consistently proc this bonus damage. Her E grants a basic attack reset, letting you E and instantly smack them with the proc. Overall, must build, every game.
Rylai's Crystal Scepter
The bonus health is a proxy for getting Warmog's, but this item grants Katarina utility in being able to slow her targets and the entire time. Also grants her AP to crank her damage up. If anyone can find a replacement for this item, I'll consider it, but for now I believe this to be the most viable second chance.
Ionian Boots of Lucidity
Grants you +10% cooldown reduction, which is pretty important on this build to keep you spinning for longer.
Warmog's Armor
Massive amount of health, cooldown reduction, and that passive has insane synergy with just how well she can escape and come back into a fight.
Honorable Mentions
Stuff that I believe is particularly effective. Recommendations will be considered for this section! Titanic Hydra
I usually buy it as a 5/6th item, it's pretty costly and you need to get your other stuff up to scratch first. Rylai's is cheaper, and probly has a more noticeable effect than this early game. Late game however, this mixed in with a Trinity proc can be a scary thing to be punched with. You'll probably have 4k HP at that point, so you'll be dealing around 400 bonus damage when activates. You can optionally purchase this second if you would rather it over Rylai's.
Thornmail
The passive is alright, but the armor boost is outstanding. I've been throwing this in as a 5th item against AD heavy teams, works wonders.
Her Arsenal of Stuff
Seriously SO many items can have synergism with this build, granting it huge flexibility. Is there a Warwick on the enemy team? Grab a Last Whisper. Want more burst? Grab a Luden's Tempest or Rabadon's Deathcap. You can pick anything you think warrants to the situation, more tank stats, more damage whatever is needed. I can't give you any set choice to finish off your build.

Voracity
Resets your abilities on kills or assists. It's useful, but not as much as on AP Katarina.
Steel Lotus
When you pick up a dagger on the ground from her Q or W, you activate this ability dealing a large chunk of AOE damage. Activating this also drops the Cooldown on your Shunpo by a massive amount, so make sure you're setting this off as often as possible. It should also be mentioned, that the cast point of this ability, is from where Katarina picks up the dagger, NOT from the center of the dagger. Where you pick up the dagger really can make or break a trade. Bouncing Blades
This goes on a really low cool down once you've built up your Cooldown Reduction. Constantly flick these out in trades and during teamfights. Drops a dagger behind the target too, which you can use to poke the opponent or keep as a threat/escape fallback.
Preparation
Grants you a movement speed boost which is useful for catching up or escaping. Drops a dagger where you cast this ability after a short delay too.
Shunpo
This is the fun of Katarina. Probably the only reason I play her. You get to jump around like an absolute mad(wo)man. Make sure you have quickcast set on as well, because it positions according to where your cursor is pointed when you cast it on the target. It can be used to chase, escape, dodge, and to generally mess with your opponent. Also resets your basic attack timer. FUNFACT: If you E onto a blastcone, Katarina will instantly attack it from where she E'd from. Damn cones have ridiculous hitboxes.
Death Lotus
I mean it is your ult, but it does wwaayyyy less damage when you aren't building straight AP (Or AD for you absolute crazy ones out there). If you're out of abilities, or just jumped into the middle of a crowded fight, drops this down and try to jump at a low health opponent ASAP, because you'll do more damage using a trinity force proc combo rather than waiting for this to finish. Has really good synergy with Expose Weakness too, boosting your teams damage by 3% against anyone hit by this. Since you build tank, you can't just be clicked and instantly made irrelevant by let's say... A Lissandra for example.
Combos
I wouldn't really say there is a set combo for Tankarina honestly, but it's really using your abilities to avoid damage as long as possible, while putting out as much damage as possible at the same. If you can Shunpo to a dagger, using Preparation to get away from the opponent, then Shunpo back behind them when they come to chase you, you've won the trade and have a spare Shunpo to keep going in or escape. If you have the advantage, or are confident you can win a straight trade, Shunpo onto them, drop Preparation, and then Bouncing Blades them as soon as you toss up a W. This way, they have as little reaction time as possible against you. If you open up with Bouncing Blades they will expect you to jump onto them and might be ready for it. It's all about being as unpredictable as possible. During all of this of course, you should be procing your Trinity Force off as often as possible, as that really punches into an opponent when it hits. With such an in and out playstyle, it is entirely realistic to expect getting a minimum of two off in most trades.Chasing
Chasing on this build is pretty insane. With 40% cooldown reduction (Which you usually get with the core build), your Bouncing Blades goes to about 4 seconds while Shunpo goes to about 5. When I start chasing someone I generally use E W Q. The E will slow them from Rylai's Crystal Scepter holding them in placed for your W. Usually, the dagger will bounce in the direction they are travelling, making for an easy followup. If they escape from your W, you can still get the reset and follow up on the Q. With such low cooldowns, you can constantly repeat this pattern over and over again while hitting with Trinity Force.Escaping
Make sure you put wards and E onto quick cast. This means, with enough practice, you can escape anything if your reaction times are quick enougREEEE RIOT. Escaping on new Katarina is fairly difficult outside of a teamfight. Try to Shunpo towards minions if there are any available, or use Bouncing Blades to land a dagger where you want it to be. Preparation is a move burst.Be unpredictable
This comes with practice really and applies to every champion. Maybe Shunpo away from an enemy and use Preparation to have them foolishly try to chase you right ontop of a dagger. If your opponent can't foresee what is about to happen, you have an advantage over them that they don't have on you.

Early Game
During the early game, it's mostly about surviving and making small skirmishes off with your opponent. Farm as much as possible to get your Trinity Force complete. Help your allies if you can. Most opponents don't predict Trinity Force Katarina and don't know how she works, making her a mean surprise when she starts rolling.Mid Game
Mid game is when you can really become a monster. You should have Trinity Force by this stage and some other items that boost you tankiness. You can follow targets exceptionally well, you can split push effectively. You can't really act as a "soak" for your team yet, since you lack the tankiness to stand between multiple opponents, but you can definitely swallow the combos of champions with fairly high burst. This is also a really good fear point for you enemies. Your damage and Tankiness at this point in the game can really tilt opponents too. Keep the pressure on at this stage because this is probably your strongest point. If you are fed at this point, you are a nightmare. I have fond memories of killing a team over and over again in their base, for me to just heal back all the damage with a Warmog's and repeat.Late Game
You stop being such a huge threat at this point of the game. People have lifesteal and the resources to deal with you. Stick by your team, and the opponent. You don't have ridiculous instant kill burst, but you do enough damage that you can't be ignored by the enemy team. If you reach this point of the game, it's up to how your team capitalizes on the opponents trying to fend you off. That's not to say you can't fight anyone, you can still pump out a fair amount of damage if someone tries to challenge you, but you definitely aren't a massive unstoppable force.

I'd really like if anyone else has any thoughts on this build, because I'm constantly looking for ways to make it as strong as possible!