Pros:
+ High burst damage late game
+ Shunpo doesn't really do that much damage now but it's still a good escape tool
+ Ultimate targets more than one champion
+ Great damage output with Q
+ Super Strong farmer late game
+ Passive makes her really good at burst damage

Cons:
- Weak early game damage but builds up
- Her ultimate can be interrupted by CC
- Squishy but if you build a bit of health you will have sustain
- Mostly focused in team fights
- Melee AP caster, means that she will be poked by ranged AP caster like Lux or cassiopeia.

Greater Marks-
Ability Power early game will help you deal more damage and sustain for your squishiness. You could substitute them for Attack Damage Marks since it will increase your poke in Melee range, the damage on your W (it does scale on AD) and your last hitting but I prefer using Magic.Pen Marks.

Greater Seal
More Ability power will mean you can deal much damage early game and more late game. Also it helps you farm with your Q early game.

Greater Glyph
The Flat AP ones will help you if you wanna be aggressive and go for kills.

Greater Quintessence
For Kat i pick Spell Vamp Quintessences because i feel like she get pushed out of lane really quickly and easily and once she gets a Hextech Revolver she will have lots of sustain and damage.

Flash - Helps you make plays, escape, position yourself for your Ultimate. This Summoner Spell is a must for Katarina

Ignite - Helps you in killing your enemies faster. One thing to note is that you can cast ignite while using Death Lotus so if someone barely escapes but you tag him with ignite and he escapes on really low health you'll probably grab the kill unless he gets really lucky and survives with 1-10 hp.

Alternatives:

Exhaust - Good for easier 1v1 duels. Also good for shutting down someone in a team fight. I don't use it a lot but it's good.

Heal - I'm not a big fan of heal but I guess it might work on Katarina if you use it before you get ignited. Still..I wouldn't recommend this in ranked games but in normal games...