This build lets you be fairly tanky and also do extremely much damage. It allows you to jump in fights you will 100% die in and then just walk away (or kill everyone)after getting the carry.
All the items synergize and allow you to do things you can't do with AP.

You can build it into a full AD comp and still do an insane amount of magic damage.

BOTRK lets you be really strong early even before you build it, because raw AD is better than raw AP. After you finish BORK, you go kraken slayer second item. (don't go kraken first, because it sucks as a first item after 14.10).
The synergy between the 2 items: BORK gets them low, which allows the new kraken passive to deal more damage.

Terminus: Terminus is one of the best items for katarina (if not the best). It synergizes so well with her kit due to the fact it gives you both armor pen and magic pen. Even though you build AD, your daggers and abilities still do magic damage. (that's why terminus let's you blow up people)

Terminus also lets you become a lot more tanky without building flat HP because of the other passive. Your ultimate procs the item so fast and allows you to just jump in 3v1's and barely take damage on your third item.

After temrminus, we must build another item to help us remain tanky and still do damage. That's when titanic hydra comes. The item itself doesn't really contribute to the tankyness except with the HP, but the strong part about it is the active.
Your combo with titanic:
auto-titanic-E-auto

(WARNING: DON'T USE THE ACTIVE WHILE ULTING OR IT WILL CANCEL THE ULT)

Now that you're tanky enough, you must get the final item that lets you oneshot even the fed tanks. Rageblade. Rageblade applies every single on-hit effect in your inventory twice on every fourth auto attack, which makes both your autos and ultimate oblitirate people.

Level 1 - Try to get the first 3 melee minions with your Q and don't lose any HP.

Level 2 - 90% of the time, Q then E and spam auto attacks to either get them low or even score first blood (if the enemy is squishy enough). Use your auto E auto reset. Don't waste your summs level 2 because the enemy can easily flash away and you don't have W or E to follow. DON'T TRY THAT IF THE CHAMP IS STRONG LEVEL 2 (yasuo, yone, talon, etc.)

Level 3 - If you didn't score a kill/force a recall then, now's the time. Q a minion, E then W to follow. Pick up the dagger and auto attack until you kill them.
Remember to combo right and to use your auto E auto reset.

Level 4 and 5 - You can keep with the same strategy, try to farm as much as you can without getting poked down and punish the enemy for cooldowns/creeps. Always help your jungler and even encourage him to fight in the river. You win all the early skirmishes if played right. Ping and communicate.

Level 6 - You've got your ulti now. Don't waste it trying to get a solo kill because the enemy will just flash away and you will miss your ult.
Do that only if you're sure you can get the kill (for example if the enemy is low) and then proceed to all in.
If you don't have the ability to get a solokill at the moment, shove the wave trying not to lose HP (Q-E-W run away E onto the dagger to clear waves safely) and then go to roam bot/top.
Only roam to top if the enemy toplaner is low and you are sure you can 1v1 him (toplaners can and will get double kills on roams if they're strong).
Prioritize botlane. Just walk up there, E W onto the adc/support and try to blow up the one on less HP first. Chase down with Q and E.

If you can't roam because the lanes are pushed in, just shove the wave and recall. Try to get pickaxe/vamp sceptre your first back. You can also go for a dark seal first back if you want a little bit more HP and AP.

Level 6+ - Keep on with the same strategy. Look for roams/skirmishes to go in and just focus the right targets. Once you get your BOTRK you can 1v1 almost anyone and completely snowball from there.
Kraken slayer is a huge powerspike because it allows your ultimate to melt people down even faster without you having to chase. Really good in 3v3s.

Trading & No ult combos:

auto E auto - auto attack reset, you must use that all the time

E W Q autos - engage combo that allows you to chase after them with the Q. you can delay the Q and hit it in the direction they're going

Q E W autos - trading combo in lane, mostly used when you don't plan on chasing

Ult combo:

E W R (wait) Q E - never cast Q before your ultimate because then your dagger is easily dodgeable and the enemy has an easy escape. always wait for the enemy to start walking away from your ult, then Q and immediately E (your W would've resetted your E by now)

Q E R (wait) E W - similar to the other one but you engage with the Q and follow up with your E

Titanic hydra:
auto titanic E auto - burst down enemies

- Don't overthink the combos too much.
You don't have to memorize each one, you just have to have an idea of how your abilities work and how you can synergize them and you will make the combos on your own. You must always get an E reset and have the ability to chase the enemy, that's all you need to know.

- Don't take risks in lane
Simply don't all in people and waste summs, etc. out of greed or tilt. Short trade (or do whatever is appropriate in the matchup) and play strategically. You should play around your team and not give the enemy free gold for no reason.

- Be patient
You must be patient when you play kata. That doesn't mean you have to sit for a lot of time waiting, you must be just a little patient and understand when you have to go in.

- Play her a lot
Kata can't be learnt with reading. No matter what you read, how much of it you read and etc. you won't learn her unless you spam games.
Try to play her in normals, pick her into counters and just limit test yourself until you're comfortable on her. Skill comes with experience, you will learn her just by playing.