You will pawn from the start, is what katarina needs or you will become a feeder and that's not good. With this build you will start with good AP to make a first kill or maybe get more than 1 kill in the gank, and when minute 15 comes they will surrender if they are intelligent enough for not wasting they're time dying because you will be killing them like flies lol.

There are runes with flat ap and scale ap, scale runes give better stats at the end game but I using flat AP runes you have better stats at the beggining and katarina needs to pwn from the start, plus you have the extra AP from masteries giving you a total of 25 ap without AP items at the start, more than an amplifying tome, also good magic penetration from runes and masteries. Your shunpo will make good damage. HP runes gives good survivability to katarina, 48 from runes, plus 48 from masteries and 120 from doran's shield a total of 216 HP from the start. Good enough for pushing as kat and getting first blood.

I think masteries are so obvious for Katarina, so I will not explain them

Almost all your items are from the recommended section, but the trick here is what items took first. I always start with doran's shield I think is a most for katarina, I have seen katarinas that start with boots and potions but they don't have the defence and health from doran's shield and die so quickly, as Kat you can't die also I don't recommend starting with Amplifying tome you already have good AP start, you could buy it and have 45 AP but you will not have 120 health from doran's and the armor that suits well katarina. The next item you will buy are basic boots, after you get boots you will concentrate in hextech this item is what makes Katarina a true killer, is a key item for Kat, with the active from this item you will surely get a kill, plus you have mor AP, more AD, Life Steal, Spell Vamp the combo is Killer instinct, hextech acitve, shunpo, dead lotus, after getting hextech you will get magic penetration boots sometimes I get rylai's or mejai's just depends if your team is doing great get mejai's if not i will not recommend getting it. By this time probably the other team start the surrender vote if not get the next items and kill them all. You also want to use your own judgement for what items you will need, but never never forget Hextech, I think is the main Item for katarina, if your fighting against a tank team, void staff and abyssal scepter are good, also abyssal is really good if your against AP team for the 57 magic resistance. If you are facing Champs that are disabling your ultimate, really consider buying vanshee's veil

Shunpo is your main damage skill, is the first skill I always get and it's so good for the first gank you can shunpo and use flash to escape if they target you also if you or a teammate get the first blood your shunpo will be ready for use it on another champ, is also the first skill to master.

with exhaust and hextech basically you can fight against two or maybe thee champs and kill them all you exhaust one champ and use hextech active in other one shunpo and dead lotus, also is really good having exhaust because you have a good way to slowdown a champ before you get hextech. Flash and shunpo are a good combo to escape, chase, etc.

You will farm with champs. I think is the way to farm with katarina, she is a real killer and champs will make you rich, don't be afraid to push as katarina, with kat you can also kill 2 or the 3 champs in front of their turret and running away before turret kill you with shunpo or flash. Use your judgment when you can beat 3 of them by yourself or when they can disable you and kill you. When killing minions spam all your abilites, bouncing blades, shunpo, cooldowns are really low and you don't need any mana with kat.

Sometimes is hard to pawn with katarina, that depends almost on what champs are you facing up, in my personal experience as katarina you will have problems with Xin his stuns cancel your dead lotus so easy, when you are facing this kind of champs that disable your ultimate is a good idea to wait and be the last champ in your crew to jump into fight. Consider that it is also important how well are your teammates doing, because if some of your teammates start feeding another champs will be hard to win.