Any tips on how to get through the laning phase as Kat?

As Katarina? Simple, honestly. Run Domination as your main runepage and sorcery as secondary, Electrocution obv as your main keystone spell. You are just gonna run typical assassin builds on the runes, aka Sudden impact, Eyeball collection and either relentless hunter, or for Katarina specifically, ravenous hunter.

Secondary for Sorcery you take scorch and Orb for obvious reasons.

Start and 3 potions, Rush gunblade starting with the AD components for a better csing with the lifeleach.

Summoner spells are and either or for insecurities

As for your playstyle, poke a lot with your Q. try to make at least one of the 3 blade bounces hit the champion you fight. Then try to use your Shunpo to land on the champion, but angle yourself close to the dagger landing from your Q so you can hit them with the proc and if you do that fast enough you will trigger electrocution. Right after using E press W for a second trigger.

You can also use your daggers as zoning material. People won't risk trading with a Katarina because of how much they are scared to get all outer by her, so people will generally avoid them as much as possible. you can pressure people into retreating and thus push a wave or freeze it effectively.

Skill builds you obv max Q first, then for better escapes in pinchy situations or against non-gapclosing but movement speed steroiding comps max W second. otherwise max E second for the reduced cooldown for your shunpo.

Roam a lot. Katarina is good for catching off guard champions or to lock down lanes that push too hard, which is gonna be botlane most of the times.

As for items, as i said rush and then choose between the order, but generally buy These items:

You can swap a Lyandris for a Rylais if you want, but Lyandris generally proves better with the added M pen and the passive damage it gives.

Usually rush rabadons as your 2nd last item and voidstaff being your last one.

I hope this helps you out somewhat

You don't even need laning phase...

This is actually pretty darn good. Thanks a bunch!

Maxing order: R->Q->E->W

Runes: Electrocute, Sudden Impact, Eyeball Collection/Zombie Ward, Ravenous Hunter/Relentless Hunter. Triumph, Coup de Grace OR Nullifying Orb, Scorch

Starting items: Dark Seal/Doran's Shield/Long Sword

How to lane as Kat:

To keep it nice and simple, know what your opponent does (their champion) and know the amount of damage Kat can do at levels 2-6. As a side note laning is honestly not going to be your biggest focus on Kat, as bot lane roams are.

If your opponent plays back or does more damage than you don't bother fighting them until levels 3-6. If your opponent overextends and you're not disadvantaged significantly go for an all-in. Only engage on opponents if you're confident.

Note: If you really want to get through the laning phase of Kat just don't fight your lane opponent at all unless you know how to. Kat is decently hard to gank due to Shunpo and the movespeed from W, so you're pretty safe there. But if you choose this option you are missing out on how ok she is in lane.

Whatever you do, don't pick her into Kassadin.

You win lanes hard as fuck. What are you talking about.