Hi there! My name is Jez Stukeley(props if you know the book :D). I'm a platinum bot lane main. I usually play typical bot lane champions, but if I see good synergy of counter, I try to play something.... Untypical.
I picked Katarina support once, by an accident(I typed 'tar' into the box in champion select, and I picked Katarina instead of Taric :P). And I REALLY enjoyed the game. After that, I played about 40 Katarina support games with and against different champions.
"BUT WAIT, UNCLE JEZ, KATARINA IS NOT A SUPPORT CHAMPION!" - typical you.
I hope this guide will help you understanding the hidden power of Katarina in the bot lane! ;)

PS: Sorry for my poor English, it's not my native language. I'll try to do my best ;)
PS2: Sorry for the ugly graphics, but I'm pretty sure the guide will be better readable with them. ;c
PS3: It's my first guide ever; I'll try hard to make it perfect, but don't judge me too hard :P

PROS:

-great poke in lane
-level 6 powerspike
-an escape skill
-can carry games even BETTER than Katarina mid(in the bot lane you can feed both your ADC and yourself)
-can do a lot of things even if behind

CONS:

-no peel for the ADC
-no CC(Rylai's kinda fixes this)
-accidentally stealing minions with Q while poking(jk, all planned)
-weak against Sona ;_;
-very squishy

My favourite ones are: Magic Penentration reds, Armor yellows, MR/scalling MR blues and AP quints. You can replace 4 of your Armor yellows with HP ones, as well as your blues with flat AP OR, even better, with 6x CDR and 3x flat MR.

Amor(HP) and MR will help you survive the early game, AP quints will make your poke even better, and penetrarion marks will increase your damage in every game phase. If you get CDR runes, they'll increase your early poke.

12-18-0 <-- this is the best choice, mostly because of other keystone masteries nerfs.

For items, you always start with Spellthief's Edge. Why? Because it gives you a lot of gold and increases your overall damage. Then, you upgrade it to Frostfang as soon as possible.
Typical build: Your core items are Eye of the Watchers and Luden's Tempest, you must get them as soon as possible. Since Green Wards have been removed from the store, you should buy a Sightstone, and fusing it into Eye of the Watchers saves you an item slot. Then, you need to get Rylai's Crystal Scepter. After that, go for Rabadon's Deathcap, Zhonya's Hourglass, Abyssal Mask, Void Staff, depending on the enemy team. Don't forget to upgrade your boots to Sorcerer's Shoes!
TL;DR: > > > > >

You max Q first, Katarina in the bot lane must poke hard. Maxing your W is pointless, 'cause you can't use it very often in a 2v2 lane.
But why would I like to max E second? Well, it's a preference. I really like it, because, for example, if you go on someone, kill him, jump away, and enemies still chase you, the cooldown is lower. It's full 6 seconds difference between 1st and 5th level of this skill! W's damage is similar, but E is a point-and-click skill. Mostly because of this, I'd choose E second. But, as I said before, it's just a preference, and maxing W second is good as well.

Why do you get E at 2nd level? Because with that you can Q, wait a second and then E on enemy if you feel you're in the winning position. Especially if you play with Kalista and she gets W second, and you pop the mark. It's gonna hurt! Also, I've seen many '2lvl bot ganks', so E can let you escape from them.
> > > sinister steel

There are 2 ways: Flash + Ignite or Flash + Exhaust.
Both of them are good on Kat support, I personally like Exhaust more than Ignite. But, Exhaust on Katarina supp doesn't work like Exhaust on every other support. As Katarina, you don't want to use it defensively (exhaust enemy assassin, so they deal -40% damage, to protect your adc). You want to exhaust enemy carry, so they can't escape from your burst.
It's basically a preference, you can get any of them in every situation.
+ OR +

As I said many times before, you must poke, poke and poke. If no enemy is in your range, try to Q a minion, so it can bounce, hit enemies and you get bonus gold for that. Try hard not to steal minions with your Q bounce. If you see a good opportunity, you can jump on enemy champions, pop the Q mark and deal some bonus damage. And that's all you do before 6.
As soon as you and your AD carry hit 6th level, you can really think about scoring some kills(level 6 powerspike! ;D). Make sure enemy team cannot interrupt your ultimate, wait for jungler(or for adc to be ready) and just kill them. Do not care about stealing kills(let ADC pick them up IF YOU CAN); once a wise man, an adc main said: "You say 'sorry' for taking a kill only if your adc does not farm; 1 kill = 15 cs, so where's the problem?". As soon as you get Luden's Echo, your poke becomes even stronger. Important thing: do not forget to buy wards(try to have 2 in your inventory at a given time), potions and upgrade your trinket!

Katarina best synergies:

Ashe - Tons of CC plus a good poke from W make you easier to destroy enemy bot lane.

Caitlyn - Great poke connected with yours makes enemy bot lane CRY YOU A RIVEN. If an enemy accidentally puts his leg into Cait's trap... Well. RIP.

Jinx - Good poke, some CC and a global range finisher. Works really really well.

Kalista - One of the BEST synergies with Katarina. Tell her to get W at 2nd level, then simply Q+E+autoattack an enemy. If Kalista autoattacks him as well, it's gonna hurt. Also, Kalista's ultimate. When she ults you into enemy team, you can instantly start spinning while they are CC'd. PLUS she has got a lot of damage.

Annie - It's not a "typical" bot lane, but it works really well. Annie's poke, stunning passive and great engage make her a great ally for Katarina. But how to make a good teamcomp with Annie + Kata bot lane? ADC top lane? Tanky jungle, poke mid? Good luck at thinking about it! ;)

Thank you so much guys for looking into my guide! I really tried to do my best to show you the hidden potential of Katarina support. Remember, it's still Work In Progress; I'll update this guide to make it prettier and better. Thank you once again! Please, leave a comment if you can, so I will know what to change here. Love you all <3