Not Updated For Current Season
This build is primarily if you're in the mood to carry. As you can see its obviously for Katarina, so carrying won't be all that hard. The reason for the name of this build is because you actually build Lichbane. I know it sounds weird, but it surely pays off later. Since there is an immediate auto attack after you Shunpo, you are guaranteed to have lichbane proc, plus it makes farming and killing super minions or jungle creeps super easy and effective.
P.S. This build is for mainly Mid and late game, but since you're playing Katarina, ganks at beginning game can be easily achieved, and will help you build Mejai's stacks a lot easier.
To start off Katabane, you will want to have a primary AP runepage. Many people have this runepage, but with a slight modification you can carry your team to victory.
I first start off with:
Marks: Straight Magic Penetration Marks
Seals: Straight AP Seals
Glyphs: Straight CDR (Cool-down Reduction) Glyphs, and lastly:
Quintessences: Pure 4.95 AP Quints.
This will give you a good advantage in early game ganks with starter AP, and will also help you slice through MR tanks.
The masteries for Katabane can be spread out to your liking. When I play, since I'm lazy, I usually just go standard 21/0/9 Build with her, picking up increased magic damage, and also the AP per level.
If you want to change this, you can either go 9/0/21 and pick up Cool-Down Reduction and swap out the Glyphs and change it for more AP. Its entirely up to you when it comes to the masteries, but I usually avoid too much into Utility since Katarina has no mana base.
This is my favorite part : killing people.
There is no set max damage skill order that I currently know about, but when going into battle you want to target the squishiest character there, and open with:
Bouncing Blades > Killing Spree > Shunpo > Death Lotus > Shunpo > Bouncing Blade > Rinse & Repeat.
You can change this to either Shunpo then Bouncing blades, and even swap the order you open with Killing spree depending on if it's a gank or a Teamfight.