if have been playing karthus for a long time and this is the most successfull build i test.
Suicide Karthus is a suicide karthus, who suicide to make a pentakill suicide.
deals tons of damage and is easy to play. This guide consider non-tankieness! but hey! i dont care 1 death for 5 kills. good deal!

Death Defied: Upon dying, Karthus enters a spirit form allowing him to continue casting spells while dead for 7 seconds.
-very nice passive. in ghost state you can continue spamming your skills without manacost over 7 seconds.

Lay Waste (Q): Creates a delayed blast at Karthus' cursor position. After 0.5 seconds, deals 40 / 60 / 80 / 100 / 120 (+30% of ability power) magic damage to each nearby enemy, this deals double damage if it hits only a single unit.
Cooldown 1 seconds
Cost 20 / 28 / 36 / 44 / 52mana
Range 87
-very nice skill to harass your enemy or lasthit minions earlygame and endgame

Wall of Pain (W): 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 22seconds
Cost 100mana
Range 1000
-very wide castarea. if you cast it infront of you, it should block more than the hole lane. can be cast in fog of war or brushes to reveal all terrain it is put on.

Defile (E): 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 0.5seconds
Range 550
-best skill to farm. also restore mana when you kill a unit.

Requiem (R): After channeling for 3 seconds, Karthus deals 250 / 400 / 550 (+60% of ability power) magic damage to all enemy champions (regardless of distance).
Cooldown 180 / 150 / 120 seconds
Cost 200 / 300 / 400 mana
Range 10000
-most IMBA skill ingame. deals terrible terrible damage and it's global! D:

*olololol lucker noob!! kathus R R R R R R!!?!?! fast!*
*no mana :<*
*FU -.-*
what you do? Clarity! always a good pick for Karthus. Defile also consume a lot of mana so take Clarity and it will be kay.

the other summoner spell is optional:
i tend to Teleport for much much more mobility but you can also use Flash, Ghostor Ignite

Full AP runes for max Damageoutput
Greater Mark of Magic Penetration must-have caster rune
Greater Seal of Scaling Ability Power more AP
Greater Glyph of Scaling Ability Power MORE AP!!
Greater Quintessence of Ability Power EVAN MOAR AP!!!! and early boost

Arcane Knowledge 10% MPen very important
other masteries just viable
i switch my masteries into 21/0/9 because i think u should do as much dmg as possible and u can reach it with 21 in offense and not in utility.

archangle's staff most important item in this build. each one add 3% of your mana to AP. the first archangle's staff will give you around 120AP.

Begin:

• Doran's Ring or Meki Pendant
• Tear of the Goddess to improve your manapool
• Sorcerer's Shoes
• first Archangel's Staff
• IMPORTANT: first Zhonya's Hourglass extremly important to buy this before everything else. you will live 3 seconds longer and cast your Defile 3 seconds longer! it gonna help a lot.
• than Rabadon's Deathcap
• second Archangel's Staff
  last item is viable:
• another Archangel's Staff for AP
• Void Staff if the enemy team stacks MR
  if the enemy team still dont stack MR you can also sell your boots to buy another Archangel's Staff ;)
  
  in the end you should have around 1500AP with nashor and Elixir of Brilliance

Karthus need a lot of farm and you need your Requiem asap. you should always pick mid lane. pre lvl 6 focus lasthit with Lay Waste and auto attack. tell your mate if your ulti is ready to secure kills.
so if you see:

you can:

initiate with Wall of Pain and start wreaking havoc! toggle on Defile and active Zhonya's Hourglass to put off your death. YOU GONNA DIE! but you dont care because you got your awesome passive Death Defied. spam your Lay Waste all over the enemy carries and squishies. AND DONT FORGET TO USE YOUR Requiem!!!

again stonewall008 shows us how to jungle with Karthus
tested and approved!
but not recommended for suicide karthus

try to take it as often as possible. it reduce all your skills and gives you infinity mana. just ask your friendly jungler if he possibly can give you his mana buff with a big THANKS! DUDE!

Pros:
+tons of damage
+great farm skills
+spamable DPS skills
+mass slow
+easy farm
+gather ***its and kills very easy

Cons:
-very very squishy
-first target
-rather weak endgame

zombie karthus deals terrible terrible damage and gonna be a big help for your team. you can kill squishies with a single Requiem >:D. your Lay Waste should deal around 300 per shot, your Defile should deal ~300dmg per second and your Requiem about 1500dmg.
so have fun with the SUICIDE KARTHUS!