Karthus is a range/spell character that can deal massive damage but has a small pool of hp(but it has nice base mana) and its very fragile.Karthus have unique PASSIVE and can use it wisely,killing spree after he has already died also included(with use of requiem its easily to make few awesomes kills)

Detailed information's about karthus spell/ability.
ABILITIES:
Death defied - passive This passive allowes you to cast spells for 8 seconds after your death
Best used to cast your ultimate and help your team win a teamfight. If you have the mana just toggle defile on and the enemy will drop health fast!

Lay waste Creates a delayed blast at Karthus' cursor position. After 0.5 seconds, deals 40 / 60 / 80 / 100 / 120 (+35% of ability power) magic damage to each nearby enemy, this deals double damage if it hits only a single unit.
cooldown: 1 second
Low lvl farming tool and burst damage in PvP fights

Wall of pain Creates a 800 / 900 / 1000 / 1100 / 1200 distance wide wall through target point. Reduces enemy armor and magic resistance by 15 / 20 / 25 / 30 / 35 and reduces movement speed by up to 40 / 50 / 60 / 70 / 80%; debuff lasts 5 seconds. The wall lasts 7 seconds.
cooldown: 22 seconds


Defile Toggle Off: When Karthus kills a unit, he restores 20 / 27 / 34 / 41 / 48 mana.
Toggle On: Drains 30 / 42 / 54 / 66 / 78 Mana to deal 30 / 50 / 70 / 90 / 110 (+25% of ability power) damage to nearby enemies each second.
cooldown: 0.5 seconds after toggeling off


Requiem - Ultimate After channeling for 3 seconds, Karthus deals 250 / 400 / 550 (+70% of ability power) magic damage to all enemy champions (regardless of distance).
cooldown: 180/150/120 seconds The tool to finish off enemys that get away with 10HP. It is also great to use in teamfights right before initiating. It will be a good start for your team and waiting to use your ultimate might cost you your life.

Explanation:(im playing as AP-ability power and MP-magic penetration)
GREATER MARK OF INSIGHT:this mark add's you largest amount of MP of all seal's,glyph's so i reccomend going for MP at mark slots.

GREATER SEAL OF CLARITY:going for AP seals because seal is recommended for "DEFENSE" and there is nothing i could use wisely instead of going for AP at seal slots.

GREATER GLYPH OF POTENCY:this glyph add you largest amount of AP of all marks,seals so i reccomend going for AP at glyph slots.

GREATER QUINTESSENCE OF INSIGHT:best quintessence if ur going for MP as karthus or you could also go for GREATER QUINTESSENCE OF POTENCY.

Starting with AMPLIFYING TOME means u can have some ABILITY POWER and with wisely used LAY WASTE u can easily push your opponent to his turret or back into his base.Buying HEALING POTION at start can also help u to survive some nasty gank's or stuns that would kill u, but with gank team going after u,u can sometimes save your "***" and if using FLASH u can also save yourself and smile at them.Mostly i would recommend KARTHUS playing via mid because of his ulti REQUIEM that won't let your opponent to walk free with small amount of hp and he will have to return to the base to restore hp.Buying MEJAI'S SOULSTEALER will allow u to gather stack's in early game already and with every stack u get 8 AP ( 2 stacks for a champion kill and 1 stack for an assist),this item also give's u 20% cooldown reduction if it stacks 20 stacks on it and automaticaly u get 160 AP and 20% of cooldown reduction.Only bad thing of MEJAI'S SOULSTEALER is that u cant allow yourself to get killed so take care at turrets and do not push if its not necessery.
After u get MEJAI'S SOULSTEALER u must go directly for BOOTS OF SPEED and after that for SORCERER'S SHOES because of movement speed 2 and 20 MP that's awesome(it reduces opponents MR and it allow's u to hit with your spells alot more).
After these 2 MAIN ITEMS(considering myself) u can start off with Cataclysm the protectorCATACLYSM THE PROTECTOR because of bad HP pool,after that go directly for RABADON'S DEATHCAP because of 155 AP and 30% increase of your total AP u got.(if u got 100 AP it will add u 30 AP with just his unique passive.Buying Banshee vialBANSHEE VIAL will add u some extra MR (50) and unique passive: that will absorb 1 enemy spell and will recharge in next 45 sec(mostly helping u to survive when u were absolutely sure u will die) or u can aswell get ROD OF AGES if u got good support/tank class in your team because of 80AP instead of 50 MR from Banshee vialBANSHEE VIAL(but u loose the remover).
Get VOID STAFF because of 70 AP and 40% MP(it will penetrate enemy's armor like "hot knife throught the butter" :) )
RYLAI'S CRYSTAL SCEPTER is a item that will add u 500 extra HP and 80 AP(with RABADONS DEATHCAP it's 80+24=104AP!)this item also slows single opponent for 35% of his movement speed (if used on multi targets its only 15%).

Karthus is very overpowered character that has alot of "base mana" at start,starting with summoner spells teleport/flash will mean u cant afford to waste your mana on mobs.
TELEPORT: use it wisely to help your teammates while the opposite team is pushing them back to their turret(teleport on minions behind the enemy back's and wait for them with "wall of pain".

FLASH: it can sometimes help u to initiate in fights flashing from one side to the other directly in the team fight.(plugging wall of pain/defile and with use of lay waste u can easily make penta kill-but still if u die,be aware that u got requiem rdy so u can finish off enemy summoners in first 0-4 seconds after u died).

If u think you got problems with mana pool you can also use CLARITY instead of TELEPORT but flash should stay.
CLARITY: u can also use clarity if your mana pool is drying out,when staying at mid and if ur not paying attention to your mana pool with killing minions u can easily restore your mana pool with clarity,so u can easily stay on lane(mid) and defend your turret and your LIFE!

Starting off with :
your WALL OF PAIN and DEFILE should be used first at any fight, LAY WASTE will bring your opponent down fast(if not u will 100% kill him while dead with your passive and toggling your REQUIEM "AND MOSTLY U WILL SEE PEOPLE SPAMMING U ABOUT ->"R" skill).

If needed also u can use wall of pain to slow enemy from killing you and use lay waste and defile ,that way u can easily kill them or even escape with use of flash .

BUT MOSTLY NEVER FORGET TO GET YOUR MANA BUFF!!!(adding u alot of MP5 and cooldown reduction on your spells

Regardering my build.
PROS:
-more survivability(because karthus is squishy/fragile without banshee/rylai)
-quite nice amount of AP/MP (deadly combination)
-requiem is your ultimate killing tool and watching people nerd raging is just funny :)
-great in teamfights(but be aware that u have to fight close to the teamfight because of your defile that wont toggle off even if u die(that's the thing where most players fail)

CONS:
-ur squishy even if u got items like banshee/rylai(but considering that you can be initiator also it means your team will kill all of the opponents)
-mostly karthus is not the best initiator(but with the right use of u can easily start off with wall of pain and flash back to your team while the opponents have been slowed and your team can bang them(meanwhile u can help out with lay waste and defile ).

Play Video Karthus doesn't have to be near the teamfight to do his ULTI and drag down opponents! Play Video Even when ur dead u can do your ULTI! Play Video Karthus shouldn't initiate first but when u see that your opponent's have small amount of HP u can use your ULTI and score "HIGH FIVE"!

Karthus should always watch his back because he's fragile(he mostly cant do anything again's ganks,but playing as initiator isnt so bad after all,secrefise yourself sometimes and initiate in teamfights first,u will see that wall of pain will make your team grabbing enemy before entering the "SAFE ZONE" :))
Starting via mid at start allows u to level faster than champions at "TOP" and "BOTTOM" and with leveling so fast your ulti will soon become your killing tool in game.Always try to cast your lay of waste on right place so it will deal massive damage hitting only 1 player(if it hits multiple targets it makes only 50% of damage).Try to harras players while gathering firs't 6 levels so when u get your ulti,you can easily finish off players that were also harrasing you.
Playing safe will make yourself ultimate killing tool,but in teamfights its even better.

Normaly when im playing i can easily solo 3 players because of stack's i got from Mejai's Soulstealer but some of players also got the "OH" skill for karthus and that's SILENCE.
When ur silenced u cant even use your flash,so never underestimate your opposite team and just hold your line for backup and team fight later.

I hope i will help some of player's to improve their gameplay because im enjoying this build and normaly score around 20/5/20 :)(always top score in everygame i played with karthus).

If u want me to show u how i do it ingame u can also add me ----> Biuse.
Following this guide will get u to the top :).