This is basicly how I play katarina, it works really well and i have seen so many bad katarina players with messed up builds and I figured i would make a guide.

Pros:

Huge damage from ult.
Able to shunpo friendlies, wards etc.
Her frickin' passive :D
Her ult can only be used against enemies, making it a enemy detector. It will light up when enemies are in ult range (even stealthed ones!)

Cons:

Pretty much useless when skills are on CD.
Can get screwed by stuns.
Not that bulky.
Not a very strong mob-killer. Her Q is the only AoE skill she can use against mobs.

I originally ended up with flat armor yellows and flat magic Resist blues because that's what I had, and I like to use all my hard earned IP on champions^^ It works for me so I guess it will work for you aswell, you can go flat or per lvl ability power runes aswell or Cooldown Reduction blues If you wan't more damage or more CDR.

As with the runes, AP because damage feeds my needs, and some armor and magic resist because you can't kill shiet when you are dead^^

Item build. The first thing I want to say is that early spell vamp is da shiet. Ult + spellvamp = megaheals which keeps you alive when you jump into their team and starts the soul-munching.

You can go for a mejai's after the revolver or rabadons but it might be risky, although I have been in the situation many times where I wished I grabbed an early Mejai's.

After I get spellvamp I get rylai's because of the slow, which keeps them in your ult range for a longer time. Duh.

I follow up with gunblade, The reason I choose Gunblade over Will of the Ancients is that it will give you more sustain because it cranks up your Q and R aswell as your auto attacks. While WotA only affects your ult and shunpo. Also the passive of Gunblade is great as katarina is pretty useless without her skills :P

You can allso go with a Lichbane but I don't really find it to be nessesary because of katarinas passive.

You can allso go with Rabadons before gunblade but people tend to focus you so the extra lifesteal and spellvamp will come in handy.

Abyssal sceptre and Zhonyas. If you have problems with AD go zhonyas early, same with abyssal sceptre. I would recommend to go with abyssal sceptre even if you don't really have too much problem with AP because of the MR reduction of everybody near you which is great in teamfights. if you have no problem with AD at all you can drop zhonyas and buy a Void staff if they start stacking MR.

Well, the reason I choose to go for surge is that it will give you quite e lot of AP and it will make you get even more ap because of your Rabadons passive. And doing more damage than your enemies expect is allways good. Flash? it's flash.

I usually play as an initiator with this build. The thing is that you gotta be really careful about hard CCs: Stuns, knockbacks, knockups. Anything that can interrupt your ult. So you need to be careful about when you jump in and start your spinning of DOOM.

One trick I have is to use your Q right as they get into the max range of it. That last dagger is sweet when it gets you your kill.

Smartcast! If you don't know what smartcast is then Ill give you a brief introduction.
Smartcast is that you have keybinds that skips the aim part of the skill activation. So instead of getting your "target phase" after activating your skill, the skill will be instantly activated at your cursor. This makes things go much faster. I have my Smarcast bound to 1,2,3,4; this way i can easily mix between regular- and smart cast.

Keybind your items! Keybinding your items is a must. I have my items bound to x,c,v,b,n; my recall bound to H, my ping bound to G and my "fallbackping" to T.

And Last but definiently not least, REMEMBER TO USE THE ACTIVE ON YOUR ITEMS. This is mostly to myself, I have died so many times because of not using my gunblade or zhonyas active :P

A trick with zhonyas? Katarina is relying hugely on CD so using the zhonyas ti gain some time might not be a bad idea.

Some good combo mates for teamfights: Nunu, Galio, Kennen, Wukong, Anivia (anyone who can keep them from getting away. Nunu would be really good, first he slows them, you do your massive damage and slow them even further making them unable to run away. and then they die^^