Katarina is an excellent hybrid champion, and many people don't realize this. A straight AD champion doesn't dish out as much damage as an AP champ could and costs too much, however, if you play AP Katarina, you will be almost useless when your ulti is on cooldown. This is why Riot created Hybrid Items and characters.

This build is a little on the expensive side (16,615 Gold To Be Exact), but focuses a lot on early game damage and late game push. You must remember that playing Katarina as a hybrid, means much more passive play in early game in order to do well.

Pros
High Damage Output
Powerful Multi Target Ulti
Great Harass with Bouncing Blades
3rd Biggest Tits in Game
Cons
Extremely Squishy Early-Mid Game
High Priority Target To Enemy Champions
Perverted Garens Spying On You

In my opinion, runes are meant to make up for your weaknesses in your gameplay, or to strengthen your strong points. I personally like the runes above, but you can use whatever you like and whatever suits you best.

Why Flash and Exhaust? Well, Flash allows Katarina to extend the ability of her Shunpo. Since Shunpo is a targetting skill, there are some situations where you cannot escape with shunpo and some situations where shunpo is out of range for a chase. Exhaust is an OBVIOUS spell to use for Katarina. In early game, it can be used to trap an enemy champion into your ultimate and lock them down for the kill. MAKE SURE YOU MAKE THEM WASTE FLASH BEFORE EXHAUSTING THEM FOR THE KILL.

Ignite is also a decent spell to use, but is really not that helpful in a lot of situations. There are times where you think, "wow, i shuold have carried ignite" but 90% of the time, you don't have to worry about that with the Flash Shunpo Combo as a quick finisher.