Not Updated For Current Season
When you start out as Katarina, you shouldn't have many troubles with early game. You should take Middle or Bottom, you can take top. The reason I prefer Middle and Bottom is because you need to be somewhat aggressive early game. If your team does not have a jungle you can go whatever lane. If their team has a jungle and your team does not, go top. If both teams have a jungle go wherever.
Early Game (Levels 1-6)
Because Katarina excels after level 6 you want to play somewhat defensive early game. For your first item grab a Doran's Shield for health and health regeneration. For creep kills throw a Bouncing Blade, you can also harass really well with it. If the enemy is getting aggressive make sure you can get out or your lane partner is nearby and ready to fight. Killer Instincts>Shunpo>Bouncing Blades, after that burst auto attack and use moves when they come off cooldown. If the one your facing has a heal use your Killer Instincts with Bouncing Blades for healing reduction. If you can do this you will probably get first blood. Go back and buy basic boots when you get low, unless you can afford your Sorcerer's Shoes.
Mid Game (7-13)
During mid game you are most dangerous. You should be able to Killer Instincts > Shunpo > Death Lotus > Bouncing Blades and kill them, if not you suck, no offense. By 15 minutes into the game you should have your Giant's Belt. Start ganking when optimal. By 20 minutes in if you don't have your Rylai's you're in deep $#!7. You should have scored atleast 3 kills by level 11. Grab you Hextech Revolver as soon as possible so you can get health from your ultimate.
Late Game (14-18)
You can now probably go in 1v3 and win if your keeping the right pace. Unless they have a lot of silences you should be set up for a lot of kills. Don't **** up and think you can turret dive unless their low. In a teamfight do NOT initiate with your ultimate, wait for all of the stuns and silences to go out. Then run in and LEROY JENKINS their whole team. You will probably die but get some kills. Don't go in unless you KNOW you can either hide and ultimate or you can pull the whole thing off.
Why not get a Doran's Shield, without it she's squishy as fck. Then Sorcerer's Boots because she is all magic damage even if you are building attack damage. Rylai's Crystal Scepter so you can murder children with your ultimate and not die. Hextech for health during your ultimate and even more burst damage. Rabadon's for massive damage. Will of the Ancients for more drain from your ultimate. Moonflair for Tenacity and damage.
Flash for catching up or getting out.
Exhaust people in your ult so they take the whole duration.
-Lots of damage
-If built right you aren't squishy at all.
-Easy creep farm
-Squishy if not built right