Katarina is by far one of the best assassins in the game. Her ability to gank, chase, and multi-kill is ranked top in the game. Katarina is easy to play but difficult to master, especially at higher levels of play. She is my champion and 95% of my wins are as her. My record is about 7 wins per 1 loss and have dozens of quadra/penta kills as her at lvl 30.
If you want a strong carrier and ganker Katarina might be for you.

Pros
Gets many kills
outlanes every champion but 3
amazing escapes
can tower dive early
amazing ganker

Cons
43 champions currently have ways to stop her ultimate
Always focused first
heavily relies on ultimate
needs middle or she will fall behind
Hard w.o teamwork

Katarina can farm decently but her strenths are map control. Controlling the map and ganking. It is NEEDED that after level 6 you have someone cover mid while you start ganking or you will lose it.

Also try to keep blue buff as much as you can. CD Reduction is amazing for kat. The point of Kat is FEEDING yourself. Use your team as bait. Have the solo lane initiate and harass them to half. Then have your guy bait and Ki, shunpo, DL double kill, take tower.

Whenever katarina scores a kill or an assist all of her cooldowns refresh by 15 seconds. What this effectively does is refresh Q W AND E. So after tower diving you can shunpo to a nearby friendly or enemy miniom. On 3 kills/assists your ultimate is refreshed. This enables pentakilling :).

Teleport you to a nearby enemy or ally target. If you teleport to an enemy target you deal damage(230+.7AP). The CD gets reduced to about 5 seconds with blue buff. Since this build is mostly AP based this is a nice nuke. The key thing is, you can use this to:
Kite while CD's are refreshing
escape from ganks/death
tower diving in and out
Kite away and shunpo back for the kill when they are chasing

Passive: does increase melee damage on attacks.
Active: When activated your next shunpo will reduce damage taken for 3 seconds by 30%(lvl 5) or Bouncing blade will do 100% dmg on all bounces and reduce healing taken by 50%.

This makes you less squishy during team fights and also makes healer half as useful. Rule of thumb is AlwayS keep this active in a lane. So if a gank is coming and you shunpo away you get areudced dmg taken for a few seconds incase they get a few shots off.

Throws a blade that hits upto 6 targets for damage+.8AD+.3AP(lvl 5). With a full build it does about 450-500. Very powerful. With Shunpo and BB you do alot of dmg especially in snaller fights. BB with KI can easily take off a 5th of the other teams HP. Also this is your farming tool as well. This is also your primary early game harass. Dive a tower, throw BB and shunpo away to a minion.

Next to crowstorm this is the most powerful aoe ultimate in the game. It deals at lvl 3 85 damage+.55AD+.25AP. So with 100 AD. And 500 AP it does per dagger. That is 260x3 daggers per second. So 3 champions take 780 damage a second for 3 seconds. That's over 2k. Now you have about 55 Magic pen. So on most champions it will do over 2k.

Heres the problem. Currently there are 42 champions with atleast 1 spell that can stop it. So you have to rely on your team and positioning. Do no initiate if the other team has CC. If the other team has 0 ways of stoping you. Feel free to pentakill...

Her ultimate takes planning, you generally want to wait until you see that the cc is burned then jump in and use it. However feel free to stay back and toss a few BB first.

Marks-Magic Pen
Seals-CD reduction Flat
Glyphs-Magic Pen
Quints-HP Flat

As I've been getting higher in ranks the early game 78 HP is better then 6 more magic pen. CD reduction is the best seal due to not magic pen available. With sorc boots at level 6 you have 35 magic pen effective increasing your ultimates dmg by 30% early game

I used to run 9/0/21 but the higher up I got I realized Cleanse is needed and the extra defense is as well. Utlity or defense should be 21 not offense. Katarina has enough power she needs extra other stuff to enhance her.

Doran shield-gives katarina a tota of 720HP at lvl 1 with masteries and runes. Helps keep her in the lane and ganker earlier on

Sorc Boots-Gives needed movement speed and MUCH NEEDED magic pen for the entire game

Hextech Gunblade: every piece of thise item screams katarina. It gives her some melee damage so auto attack isn't useless, gives 20% life steal and spell vamp. Note spell vamp works o. All bounces of Q and on E and Are.

it adds:
71 damage to Q
53 damage to E
53 damage per dagger to 2(159 extra dmg per second).

Rylai- makes Q E and Are slow the enemy by 30%, 15% on R. It makes it very easy to hase people AND gives much needed HP
it adds:
24 damage to Q
56 damage to E
20 damage to R per dagger(60 per second)

Hourglass-gives MUCH NEEDED armor to help protect from being
NUKED by melee carrys. The use is amazing. If you initiate and use it, they will blow CC's and as ur team arrives it's
GG. Or if they focus you, it gives you time then ultimate. It also adds 100 AP
Which adds
30 damage to Q
70 damage to E
25 damage to Are per dagger(75 damage per second.

Abyss Scepter-This adds much needed MR. To protect you from caster nukes AND gives team wide magic pen. every useful item and adds 70 AP.
this adds:
21 damage to Q
49 damage to E
18 damage to R per dagger(54 damage a second)

Deathcap-The final touch to make it game over. This will increase your damage by about 30%. Now why get this last? Abyss and hourglass help you stay alive and continue to gank. Chances are if you get this first you will rack up kills but deaths too.

Total you get
60 AD bonus
625 bonus AP
Q adds 236 damage
E adds 438 damage
R adds 189 per dagger(567 added per second). So 567+(85x3)=822 damage per second per champion upto 3. Now this is just item damage no masteries or elixirs included. Also this is assumed they have lower than 55 MR.

I go BB first for early harass. I start to utilize using BB+shunpo then run away. They generally will retreat. MAKE THEM USE THEIR SUMMONER SKILLS. Harass them down to about 150 HP. then go for the kill. Do not go for the kill unless you have everything planned out.

I grab 1 rank of BB then max Shunpo with getting 1 rank of Ki. So basically it's R>E>Q>W.

I used to run exhaust and teleport with 9/0/21 Masteries. But I've found in higher ranked games cleanse is NEEDED due to CC and focused. So I get cleanse and exhaust. Exhaust is for early gank and to shut down hard carries. It's also funny when they are chasing you and you KI, shunpo, exhaust, DL, BB, and you had 250 HP they had 1800 and they have a rez timer and you walk away...

Your teamates that can CC must CC their teams CC. Your DL is so strong that you WILL destroy their carriers. If a fed kat, or atleast their team isn't kat, gets a full ultimate off on 3 people you give won. Shunpo/BB for a double or triple and say GG. It's that easy!

I have about 1k games played with just about a 60/40 win ratio. Of those wins 70-80% were as katarina. I bought her at lvl 9 and played her exlcusively since. I have about 630 wins and 310 losses, my record with kat is about 480 wins and about 80 losses. I REALLY SUCK with most other characters. I don't really practice with anyone else.

Katarina won't be nerfed soon. Seeing as 2/3rds of the champions can shut her down. However is built orrectly and played smart she can own reguardless. Even tho she can't ultimate her other moves are strong and an toe to toe with anyone, Just place the ultimate correctly and profit.

Looking at my recent history, norm match my best game was 55/8/17(loss lmao), worst game was 13/11/21. Best win was 41/8/21. I would say over 60% of my games in normal lv 30 matches get over 20 kills with a 3:1 KDR. I do average .73 kills a minute, and 29 take downs(kills+assists) a game. O btw kat is a kill stealer, try to avoid I'd mid game, but you kinda need it early game.