with katarina you want to start with boots and 4 health potions because the boots make it easier to buy the sorcerers boots and give you more mobility which is great poking with Katarina, and you want the 4 health potions because it gives more in lane sustain and there are no points and buying mana

in the early game you want to poke the enemy mid laner down to half health before you use every thing on them, so what you want to do is Q them then E them then W them and run of, that adds the max damage available to you with out your ulti and it also allows a safe getaway just by running back to your minions because they generally don't know what just happened

when the enemy is on about half health you can easily burst them down with your full combo giving you and almost guaranteed kill, all you need to do is Q them E them W them then ignite and ult and you half probably gotten the kill with the massive burst that katarina has.

with katarina you want to get in the middle of team fights after your tank has initiated after that you just use your full combo of QEWR and doing about 50% of your teams damage or more in that fight but you to be careful because in a team fight because if the enemy team all targets you, you will most probably die just because you are VERY squishy and is an assassin that need mobility but you lose that mobility when you have ulted.

the haunting guise is at the start of the build for early magic penetration which is amazing on katarina. these are all very helpful and amazing items on katarina and they maximize her ability to burst down enemy targets in seconds.

with Katarina you want to max your q and w at the same time but your w before your q (if that makes sense) because with katarina her abilities do maximum damage used together and your w adds the most upfront damage where your q just adds more potential to the combo making more damage on the fist blow.

with these masteries you get more upfront damage when in a fight while adding some tankiness to make the early game a bit easier against general mid laners like ahri and zed.

with the runes you want the most amount of burst damage so ability power quints and glyths give the early ability power you need where the magic pen marks give you early magic pen which is amazing with katarina. but with the seals there are really no good ones for katarina so a bit more armor in a upfront fight against an AD champion can really help.