In Laning Phase, Katarina is pretty weak. However, there are a few things you can do.
Level 1: Just focus on last hitting with Q's. Don't walk up and don't take any poke,just stay healthy.
Level 2: Grab E. If you are taking electrocute, you can trade with Q, E, auto. If you are taking conqueror, I recommend waiting to trade until level 3.
Level 3: This is a strong powerspike for Katarina. I recommend trying to trade in most matchups at level 3. The current best trading pattern (with conqueror) is with E, auto, W, auto, Q, auto, E, auto, and finish them off with ignite. Make sure you use your w movement speed to dodge any skill shot cc thrown at you. In matchups such as lissandra where they have a non skillshot cc with a range to it, try to trade with q and e, and when you e onto your dagger stand at the max range and w away from them so that they can't root you. Generally these abilities having longer cooldowns, so you can trade them again before their abilities come back up.
Level 4-5: At this point how you play the lane depends on the earlier levels and there are many things you must think about.
Things such as: Do I have a kill yet? Can I kill my laner? Should I play safe? Can I roam? Where is the enemy jungler? These things will help you determine what you want to do next.
Past level 6: If you gain a large lead (2+ kills) before level 6, then past level 6, play very aggressively as long as the enemy jungler isn't very fed. If the enemy jungler is behind or even with your jungler, then a skilled Katarina is able to 1 v 2. If you aren't confident on her, then don't try this. Just focus on roams and plays on the map. Swap to sweeper and pick up control wards to help your roams. Try to always be available to make plays on the map.
Mid Game: This is where Katarina really shines. Her 3 item powerspike on both AD and AP is very great, and she is able to consistently hard carry her team in teamfights if you play it correctly. Look for picks on overextending squishies. Try to position yourself on the flank of the enemy team so that when they engage you have free access to their backline.
Late Game: Katarina is still pretty strong late game. There are many ways to play her at this point but currently, the playstyle of waiting patiently for enemies to waste key cc abilities with long cooldowns and then jumping in to kill them and jumping out is the best strategy.

A lot of people wonder which starters to go on Katarina, so allow me to explain this to you.
When do I start dark seal?
Start Dark Seal into mages (Orianna, Viktor, Syndra, Annie, Ryze, etc) because it is difficult to solokill them early and getting kills with dark seal + electro makes it difficult for mages to play the game as they get instantly deleted when you are very fed and have electro.
when do I start long sword?
Take long sword (with conqueror only) into poke lanes where you are confident in your ability to get a solokill as the early AD really helps with extra damage and the pots are good because 2 extra pots > 80 extra hp and 4 hp per auto. You don't have to turn this long sword into anything, just keep it and sell it when necessary. Example of good lanes to take this into: Kassadin, Xerath, Veigar.
When do I start Doran's Blade?
Start Doran's Blade when the enemy midlaner wants to all in you. Can also accompany this with Bone Plating and your all in is pretty much unrivaled if you land your daggers. Examples of good lanes to take this: Yone, Yasuo, Talon, Qiyana, Irelia.
DO NOT TAKE DORAN'S BLADE INTO MAGES YOU WILL GET DESTROYED IF THEY AREN'T TERRIBLE!!!!!!

Ahri: Ahri can use her e to stop your ultimate, and her w movement speed to avoid daggers. Past level 6, it's very hard to kill her. My personal recommendation in this matchup is looking for a pre 6 lead and then going for roams after to ensure your snowballing.
Ahri doesn't have the strongest pre 6, she has low wave clear, high mana costs, and she cannot all in you from full hp without losing, so vs her I always take conqueror, as her only form of self peel is the charm, which is easily outplayed. Make sure you take long sword if you decide to go the same route as me. Let her shove you in, only use your q to last hit, don't bother trying to poke her unless she lets herself get poked when you use q for a last hit.
If she uses her w while pushed up, all in. She cannot escape and dies or flashes.
Akali: Used to be pretty hard, is beatable pre 6. You can space her well and win this lane by clicking q on her just outside the range of the ability so your character moves into range to cast and as soon as you cast the q run away to avoid her q. I take electro into her, as her base HP is 500, and you can kill her at level 2 with q e auto (electro) ignite auto e auto. Assuming you have poked her a bit at level 1 even less than that is required. Past 6 DO NOT fight her if you are even, just roam or CS with q safely as she will always win a level 6 all in if she isn't massively behind.
Anivia: Dodge her q and win. If you don't dodge it, then make sure the time you are stunned for is the time katarina is picking up her dagger, so you aren't losing any damage. Do this by using e and then auto and w and walk at her, if you stand on the edge of the dagger, you will still hit her with it. I take conqueror and long sword in this lane, as Anivia usually will take magic resist vs Katarina, and her base armor is extremely low, so you can heavily punish her early for messing up. Make sure you go for 1 mini trade though and get her to 80-70% hp because if she has full hp it is very risky to all in when she has her egg and if her jungler has eyes you'll die.
Annie: Vs Annie, it is very difficult to solokill her because her stun has no counterplay, you cannot dodge it. You can go for an all in if she didn't poke you early, but it is impossible to avoid her poke as her auto attack range is ridiculous. I take electro and resolve (bone plating + unflinching) in this lane, and I don't even bother trying to kill her. Rush Sorcs and roam as early as possible. Make sure you don't lose too many minions/exp for roaming. Katarina DOES NOT need high cs, the best Katarina players still only average around 7 per minute. You still need exp though, so despite not being able to kill her, minimize your exp lost.
Aurelion Sol: Not a matchup I play all that often. A really good Aurelion Sol is not killable as Katarina. If he takes Phase Rush, just play to roam and help your team, as Aurelion Sol is not a solocarry champion, he needs his team to be fed, similarly to Twisted Fate. Take TP and play patiently if you have to. Otherwise, you can look to kill him if he misplays and walks up to the line of the river wall.
Azir: Electro + Resolve + Doran's Shield, sustain and scale, and look to roam. If you find kills outside of mid, Azir can't do anything and he will just get stepped over.
Cassiopeia: * Cassio Ult Stuns you if you were looking at her before you cast your ult so make sure you look away before ulting*
Don't get poked early and all in her at level 2 with doran's blade and conqueror. Try to all in her when she uses her q on you and use e to dodge the q. Make sure you take flash in this lane, because if she flashes away and you don't kill her at level 2 it is VERY hard to find a lead vs her due to her w making you unable to use e or flash. If she takes exhaust there is no hope of ever killing her solo so just pray your team saves you.
Corki: Try to trade Corki whenever possible past level 1. The best time to trade him is when his E and/or W are down. Make sure he pushes you in early. Take Conqueror and long sword and ignite in this lane. It's pretty easy if you don't get poked too hard level 1. Rush Rocketbelt vs him because it helps deal with his spacing. If he walks up to r you without his dash, rocketbelt in and kill him.
Diana: Diana is currently much weaker than Katarina in all ins pre 6. Avoid her q's whenever you can and past level 1 if she tries to use q to poke you jump onto her to avoid her q and all in her. Diana players also like to use their q to last hit similar to Katarina, so on the first wave, she will probably q the 3 melee minions. DO NOT get hit by that q. If she doesn't push too hard, you can get a cheesy all in off at if you get level 2 before her or if you get level 2 and dodge her q with your e.
Ekko: Ekko will likely try to use his e to dodge your daggers. Because of this, let him walk onto a dagger, and use w and run back a little bit so that if he uses his e to try to trade you, he will have to take the damage from your w dagger. If he does trade back, all in. Similar to Diana, you can maybe get a cheesy level two all in if he doesn't push the lane hard enough.
Galio: Make sure you take conqueror in this matchup. When trading galio, let him burn aftershock before you use any abilities. Make sure you poke him with q as hard as possible level 1. Past level 3, it is very difficult to find solo kills on him on AP katarina, so if you desperately need a lead rush BOTRK and you should be good.
Heimerdinger: Similarly to anivia, just avoid his stun and you win. If you don't avoid it, you lose. This matchup is absolute CBT similar to Lucian but a little easier so take Doran's shield and second wind and just farm. Thank god this champion isn't played by many people.
Irelia: *Rush Sunderer*
All in her at 3 by dodging her e with your w. Use it to avoid the second cast. Stand opposite of her at level 1 so she can't triple q the melee's on the first wave and then q to you for fourth stack and take half your hp. If you are scared take bone plating vs her.
Kassadin: Used to be hard, not as hard anymore. You can win all ins early pretty easily. Make sure you take conqueror and long sword or doran's blade. Let him push you in and all in at level 2/3. If he takes electro/bone plating, make sure you get rid of the bone plating before going for the kill. If you need, you can also rush botrk and you obliterate him.
LeBlanc: cry yourself to sleep at night or just ban this freak. You can't cs, can't fight, and can't roam without being heavily punished.
Lissandra: You can't kill her if she takes corrupting pot/time warp with aftershock, but if she goes electro with ring and sorcery you can look to kill her if she wastes w or e. Otherwise just roam. Consider mercs vs her.
Lux: Really easy matchup, look to trade lux whenever you can and past 6 just all in her, she can't cancel your ultimate.
Malzahar: Malzahar has 19 armor at level 1. Start Doran's Blade and try to all in him at level 2. You can also trade him at level 3. Past 6, don't ult until he ults.
He is very vulnerable to you early so just let him push you in and at level 2 or 3 you can easily kill him.
Neeko: Just trade neeko when she has used her e to waveclear. This matchup is very good with electrocute. You can outplay her by using your ignite before she uses w so you can tell which is the clone.Past 6, e w q r e ignite to kill her before her ult goes off. Like any other mage, she is easily stepped over and if you find kills she can't do very much if you play well.
Qiyana: You NEED to all in qiyana, take bone plating and last stand and don't fight her until you are looking to kill her. Past level 6, similar to akali, you probably just lose the fight, as long as she isn't 0/3+. Rush Sorc's and roam, or if they have a lot of AD you can get seekers.
Rumble: Rumble will try to poke you early, just stay behind your minions. Past level 3, if his fuel is in the danger zone, don't fight him. Wait for it to lower and then trade. Doran's shield is pretty good here (with conqueror). Otherwise, go electro dark seal and hes pretty easy to kill if you roam and get kills.
Ryze: Rush Botrk if he goes phase rush, because that slow negates the phase rush speed increase. The sustain it gives is also really good because he wants to poke you a lot early and outsustain.
Seraphine: Just scale, its a skill matchup, if she walks up too far and doesn't have w you can kill her, but otherwise you just farm and roam and its the same as any other mage matchup.
Sylas: Don't level q immediately at level 1. If he trades into you level 1 with e1 e2, you can actually start e and trade him back with auto e auto. There is this really annoying bug where if you cast your ult just after his stun it will use the cooldown but you won't get any damage from it. Vs sylas you have to rush grievous wounds or you can only fight him when he has ignite because he is too slippery to kill in 1 fight usually. Take conqueror and dblade vs him.
Syndra: This lane is really rough early, she will bully you out of lane. Start doran's shield in this matchup and just try to dodge as many q's as possible. At level 3, if you guys are even hp, kat starts to win trades. Make sure you use w to avoid an e into her orbs. Can also go bone plating if you need. If you get kills from roaming she gets stepped over just like any other mage.
Talon: DO NOT fight talon early. You will always lose. The best time to fight talon is level 5. Make sure that you place wards over walls so that you can e to him if he tries to jump out. Don't trade talon with the standard e w q r e combo because that gives talon a free melee q. Place a dagger where he will walk and then e to it and kite away with w so that he can't get a melee q and so that he can't dodge your w dagger. Use the reset you get from the w dagger to dodge the 2nd strike of his W so he can't get his passive and he should be dead. Conqueror and DBlade are also really good here.
Twisted Fate: Really easy matchup in my opinion, similar to annie except TF does much less burst. Use your daggers so that when you are stunned is the time that your dagger is being picked up so that you aren't ever losing any damage. In this matchup, ignite teleport is viable because you don't really need flash as much due to his low kill pressure on you. Teleport will help contest his roams later.
Veigar: Easiest Katarina matchup, you can E out of his cage and he is very squishy. Let him push you in and then jump on him. He can't do anything.
Vel'Koz: Also a very easy matchup, same as veigar, let him push and just jump on him, he will always lose.
Viktor: Pretty rough matchup, start doran's shield and just play for roams. Buy Sorc's first. He dies easily if you get a lead.
Vladimir: Don't q vladimir directly or he can pool it and you lose your dagger. If you are going to trade him, place a dagger where he will walk to cs and jump on him. Don't trade him when he will have his empowered q or you won't win the trade.
Xerath: really easy matchup, don't stand in your wave and let him push you in, at level 3 and beyond you always win every trade with the standard e w q r e combo.
Yasuo: The Yasuo Katarina dance is my favorite and most fun matchup. Yasuo will try to dash through your daggers to avoid any damage. When you dash to the dagger, dash to the center, so he can't avoid the damage by dashing through it. Don't use your ult before he windwalls or he can avoid the entire thing. If you struggle in lane just grab boots and roam or just grab seekers and scale.
Yone: You win this matchup easily at level 2, all in him if you can. If he respects you, try to trade him when he doesn't have any stacks of q and trade him before he uses e so he can't put space between you and his returning location.
Zed: This matchup is very difficult for Katarina, if you struggle early I recommend starting cloth armor and 4 health pots, and later build a seekers with that. Try to dash to the side of Zed to dodge his q's and to make it hard for him to outplay you. Building Zhonya's first in this matchup is good as the active makes him unable to kill you.
Ziggs: simple matchup, he will just push you in, you can't really kill him, farm lane.
Zoe: She will try to poke you early, try to dodge her Q's and use w to dodge her bubble. Past 6 you win all ins.

Just practicing katarina will help you get better with her. She is hard to pick up and is a very mechanically demanding champ. Practice combos against bots and that will greatly help you learn to play her.

In all of North America, I have the 3rd most games on Katarina in high elo. On League of Graphs, I’ve also been consistently in the top 100 Katarina players in North America.