Katarina's lane phase is all about positioning herself properly so that her Q dagger is in the right spot. When you think about it, level 2 she actually has 4 abilities. Q, E, dagger, and E reset. Start Long Sword + 3 potions and run Electrocute, Sudden Impact, Eyeball Collector, Ravenous Hunter, Triumph, and Coup de Grace for your runes. As you get closer to level 2, position yourself forward so you can immediately Q your lane opponent, E to behind them (picking up the dagger and resetting your E), and then E again. Auto them down with the bonus AD from Long Sword and ignite to finish them off.
For lane phase after that, you have a lead, so as long as you can position the daggers properly around the field, you should be able to win any fight. Remember, positioning the daggers is key, and Shunpo is an instant blink, so use it to dodge abilities that would cause your death. As an example, you have a dagger behind Yasuo, he throws his Q knockup at you, you Shunpo over the q to the dagger and proceed to burst him. Meanwhile, he just missed his main damage and won't have it up again for a few seconds. Getting a kill resets your cooldowns and you have a lot of AoE, so you should be able to 2v1 with enough of a lead, should you get ganked.
If you have enough of a lead to kill your lane opponent and still be pretty healthy, use that around the map to kill your lane opponent, push in the wave, and roam, not only solidifying your lead, but also giving your teammates a lead.
As the game goes later, keep in mind your team's engage and where you can place daggers for the maximum effect. Say you have an Ornn ulting over the entire enemy team, as soon as that knockup hits, you can q e w r combo them to blow them all up really easily.