Okay, so I see a lot of Katarina guides out there, and a lot of them have the right idea, but not a lot of them do it EXACTLY right. This is an AP Katarina guide that's obviously just way better than every other guide out there. (Jokes, but really, it is)
This is my first guide so I don't know how to make it super fancy, but it isn't cluttered up with anything lengthy so it shouldn't be too boring to read through.

Pretty simple. You'll want a point in bouncing blade first, to give yourself a ranged attack right off the bat, then rank up shunpo twice, then killer instincts once. Then just get shunpo before bouncing blade before killer instincts. (Obviously get your ult when you can.)
Killer instincts doesn't really do a lot and it affects bouncing blade the same way for every rank. It prevents damage, but your focus is getting in and out as fast as you can, so it's not necessary to rank it.

For battles, obviously we've got the shunpo>gunblade>death lotus, but I'll talk for a sec about besides that.
Her passive is what separates really good kat players from the others. NEVER forget that your shunpo and bouncing blade reset when you kill someone. If the other team is fleeing, if you can catch up to one you can kill both of them.
Also, you don't want to shunpo into a team fight because you'll get focused down immediately (don't forget that death lotus actually is ranged even though for 1v1 you'll want to be as close as possible). What you wanna do is wait for a sec (ie wait for the tank or someone not you to initiate) then:
Shunpo on the weak person, then shunpo on whoever else, and keep doing that until you stop killing people, then death lotus.
Or just kinda hang out and death lotus from a short distance then shunpo and bouncing blade to finish people off. You can use your gunblade or exhaust or wait for someone to stun the team, but if they're running they're probably losing the fight anyway.

ANOTHER THING. Kat is extremely squishy but can pull off some SICK tower dives since she can teleport back out as soon as she kills her target. Just make sure you have something to anchor onto afterwards.

OKAY. THIS IS THE IMPORTANT PART.

You wanna start of with an amplifying tome and a health potion. Then get sorcerer's shoes because magic penetration increases your damage an ***load.
THE VERY NEXT THING YOU WANT TO BUY is the gunblade, starting with the revolver. THIS IS VERY IMPORTANT.
YES, THIS IS AN AP GUIDE. YES, THIS IS EXPENSIVE AND GIVES AD. BUT IT'S THE MOST IMPORTANT ITEM FOR KATARINA. Why? Because of the slow that it gives. Get it, and you can kill virtually anyone (especially squishies) within literally a couple seconds. This is pretty much the only item you really need, since your ultimate is so damn powerful in the first place. At this point, you barely need the AP it gives. All you need is the slow.
Katarina becomes VERY easy to play at this point, just shunpo in, activate gunblade, then hit death lotus. Unless they're a tank stacking magic resist, you'll be able to bring them down and get out within a couple seconds. If they are alone, or even stray just a few feet from the group, kill them.

By the way, the Rylai's sceptre slows for half as much slow and for half as much time as the gunblade. Since the only thing you're looking for in either item is the slow (obviously they both give AP, but the reason you're buying it instead of something else), it's obvious what to choose. Without the gunblade every single champion can just walk out of range rendering your ultimate practically useless.

Now, you'll probably wanna buy more than 2 items, so here they are. You can get the soulstealer if you're getting a lot of kills if you want, but supposing you don't, follow this. Start with the deathcap for its massive AP bonus. Then (you can get these in any order, really) get the Lich Bane (giving your regular attack around + 500 damage, depending on your AP)*, the hourglass (protection) and the either Rylai's sceptre for the HP (any extra CC is nice but in this case the HP is your primary concern) or the Abyssal Sceptre for the magic resist (and penetration aura, which is really nice)

Oh, if for some reason you only want AP, feel free to just get the shoes and gunblade then stack 4 Rabadon's deathcaps. I don't really reccommend it, but, hey, if you really feel confident that you won't die, it'll give you an ***load of AP. I sometimes do it. Again, Soulstealer is also an option here. Make note that even at 20 stacks it's not a whole lot better than the Deathcap but its advantage is that it's REALLY cheap in comparison.

*OH HEY COOL PROBABLY IMPORTANT. I forgot to mention this when I originally wrote this guide, but I should explain why you're getting the lich bane. You're not going to be auto-attacking many people as you want to be in and out as soon as possible so while it does stack on your damage very nicely, it's not going to be as useful as you might hope in combat. The reason you're getting it is because its passive works on buildings, allowing you to nuke down towers very quickly. You can always leave it out for another item listed here but this is definitely something to keep in mind.

You DEFINITELY want exhaust, because it'll take the place of your gunblade until you buy it. Again, you'll be able to kill almost anyone just by having them in range of your ultimate.
I then go with ignite for a little extra damage output, but you can probably choose flash/ghost or teleport if you want a bit more mobility.

Because it's apparently trendy in every guide to list your top scores and to photoshop all your best scores into one jpeg to make it seem like you never lose so you can make it look like your guide is legit and you know what you're talking about and you don't actually completely suck so people won't yell and scream and tease and pick on you for being a noob, I'm going to do that here.
Except since I'm really, really lazy, I'll just post a screenshot of my last set of games with Kat. In fact, it makes me look more legit by not faking my way to the top. Cool.

(There should be a picture embedded here, but the image button in the editor isn't working for some reason. Maybe you can see it and I can't. I dunno. Here's the link anyway.)

http://s1179.photobucket.com/albums/x388/Inters/?action=view&current=iownnoobs.jpg

So, that's essentially it. Most characters have what I like to call a 'core' item (one that you need to get every game). For Katarina, it's the gunblade. Without the slow, or with the slow from Rylai's champions can literally just walk out of your ultimate and be fine. With the gunblade, they CAN'T. This is why you want the gunblade first. If you were to only take one thing away from this guide, remember to get the gunblade as your first item.