Katarina is without a doubt one of my all time favorite champions to play in League of Legends, and has been for years. She is lethal, fast, and yet fragile. She can target and take down multiple targets very quickly, and is nearly unstoppable when fed. Building Katarina is no small task, and there are many ways to successfully do so. Like most champions, you should make game time decisions with your build, adapting to the circumstances at hand.
Katarina is typically played in the solo middle lane. She tends to start relatively slow , but quickly can become very powerful (similar to Akali). By the nature of being a melee champion, she has a hard time early on against ranged mages, and often times has to deal with a lot of harassment and spamming. Once her Q gets up a couple of levels though, this tends to slow down. And once she has her ultimate, only summoners with a death wish would be willing to get close to her without a healthy dose of Crowd Control ready and waiting to stop it.
A typical Katarina build will have a primary focus on Ability Power. Her abilities all stack very well with AP. They also stack with attack damage though, so it is worth buying at least some of both. It is also worth getting some on hit effects because her jump puts you in excellent position to take advantage of them. You can build her in somewhat of a hybrid fashion and be very effective.
Some of my favorite items to use on Kat are the Hextech Gunblade, Rylai’s Crystal Scepter (the slow helps keep enemies in your ultimate for longer), Lich Bane, and of course Rabadon’s Deathcap. On top of these, there are plenty of other items that can be devastatingly effective when used on Katarina. Just get plenty of ability power and attack damage and you will do an insane amount of damage. Get some defense if necessary, depending on the situation.
As for boots, get Sorcerer’s Boots unless the situation really demands something else.
If you want to be really risky, you can do what I suggested in my Akali build guide (link above) and do a snowball build. Build a Majai’s Soulstealer and a Sword of the Occult right away. If you get early kills and can get close to full stacks on both items, you will be unstoppable in the middle to late portions of the game. Un. Stoppable.
Regardless of how you build her, Katarina is a very fun champion to build and play. Here is a great example video of a Katarina build I think is worth checking out:
Good luck playing Kat!