Karthus has strong early so you can trade a lot and look for kills on lane. Karthus outscales most of botlane champions so doing trades 1 for 1 is worth also when are abaut to die do it in enemy wave so you will push it during passive. Also after 6 look what your teammates are doing bcs you have global ult. Ping your support when you want to all in or when you want to recall. Also you can put deep vision in enemy jungle if you crashed wave.

After you finish bot turret go mid (if you are stronger than your midlaner also if he has tp). Play with team look all the time waht they are doing.Karthus has big aoe dmg so gruping is better for him. Try to catch some free waves on side lanes if your teammates don't have tp, you can always ult if they fight. Sometimes it's better to split bcs of your ult.(there is no rule for that, every game is different and you have to decide what to do).

Karthus can't engage on his own so you have to wait for your team to engage try to avoid getting poked to 0. Sometimes it's better to ult beofre objective so enemy team can't contest it (ofc teamcomp depentent). Karthus ult is very valuable in late game so avoid fighting without it. Look on objectives timer and your ult cd sometimes it's better to keep it for important teamfight rather than getting 1 kill.

I'm karhtus otp since 2021
I have over 930K maestry points on him
Peaked master master 253LP with over 60%wr playing mainly karthus
Also i hit master with over 70%wr
my main acc:
https://www.leagueofgraphs.com/summoner/eune/Karthus%20DMG%20Hack-EUNE
my twitch where I sometimes stream in english/polish
https://www.twitch.tv/kociokwik_

1 Try to Q->aa->QQ->aa->Q when fighting
2 Kite with your Q
3 Use Q to poke when enemy adc is about to last hit minion and is locked in animation
4 Don't be afreid to die but you have to think if it will be worth it
5 In early game you can take execute under enemy tower to push 1 more wave do deny enemy minions and next wave will bounce back to you
6 Don't play on auto pilot play less game but keep focused on a game
7 Try to last hit minions with aa, sometimes it's better to crash a weave asap and miss few minions
8 Press tab often so you can see enemy items and decide if you are stronger and should fight also check for items like banshee or seaker's armguard (stop watch)
9 Try to hit some q then use ult in last secounds of passive
10 Don't be affreid to flash on enemy team during teamfights you can deal massive dmg durring your passive
11 Track enemy jungler so you know when to push
12 Play for your limits (take fights that you are sure you win and don't over force)
13 Use q and durimg animation use "w" to boost dmg and make sure you will hit it
14 You can use "q" to zone enemys and make them stay in your "e" and aa them to deah
15 Before you zhona use "q" and active item to hide animation
16 Use "w" during laneing phase to slow enemys so your support can engage on them, in late game don't be affreid to use it so your team can catch someone
17 Some skins have more visible animations than oter i would recomend not useing Infernal Karthus and Lightsbane Karthus
18 IF you are new to karthus i recomend watching some karthus content on yt sadly there isn't muche but i would recomnd Karthus-NA streamer and old EloSanta videos to Lern karthus tech
19 Try different runes/items to find your playstyle
20 If you are gonna play kartus bot show him in lobby so your team can adapt draft to that
21 Ban your most hated matchup for me it's Jinx
22 If you want to climb faster find duo bcs playing with naut/pyke makes karthus twice stronger