Katarina is a burst damage assassin. Her main threat is her ultimate (Death Lotus). I think Katarina has one of the best passives in the game. After killing or assisting in an enemy death she gains 25g and 15 seconds shaved off of all her cooldowns. The main thing in her passive is the cooldown reduction, it enables her to ultimate 2 or more times in one teamfight.

The runes I use are for maximum early game damage. If you would prefer using runes for sustaining a lane, or improving her tankyness then go right ahead. I like to use my full AP runes because it adds on so much damage which allows me to get a kill early game which can then lead to late game carrying.

I always start out with a doran's shield because I think it is best for laning and the AP runes I use make up for me not starting with an AP item. The alternative starting items I would consider is Boots of Speed just for the early mobility or Amplifying Tomb if you plan on rushing into Mejai's Soulstealer or the Hextech Revolver.

I like to use Bouncing Blades for farming and harassment. Then when they look approachable for the kill I use Killer Instinct, Shunpo, and then Death Lotus. When using your shunpo early game, try to shunpo to them and then run towards their turret a little and then ult. The reason for this is when they run away, they're going to run towards their turret and if you're there then they run into your Death Lotus. If they see that and run in a different direction, you can shunpo to them and finish them off.

The summoner spells I use are mainly for my playstyle. I'm an aggressive player so I like to turret dive. It makes it very easy for me to turret an enemy with barely any hp when I even have 300 hp. All i need to do is shunpo on them and flash out or if they have more hp than that I shunpo on them, ignite, and then flash out. I can also see exhaust or teleport used if it is your preference.