I will start of by saying that all you need to build a pure AP carry with karthus is a lot of mana!! not too much but enough. so my build will showcase how to build karthus semi tank with a lot of mana regen and straight mana and life. Along with this you wil also have over 800AP!!!!!
For my Rune choice i take the following
http://www.mobafire.com/league-of-legends/rune-page-planner#&rune=351:9-255:9-309:9-338:39 ability per level blue tier 3
-- last hit with your "q" and you do not need mana regen
9 ability power per level yellow tier 3
-- last hit with your "q" and no need for mana regen
9 magic pen red tier 3
-- magic pen is a must early game
3 Quints of straigt ability power
-- more base damage to start
All of these runes scale very well with karthus especially late game to add to his large amount of ap late game
YOU ABSOLUTELY MUST BUILD A CATALYST OF THE PROTECTOR BEFORE BUYUING BOOTS.
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--- IT IS A MUST FOR KARTHUS IN ORDER TO STAY MID AND YOU CAN ACHIEVE THE MONEY NEED RATHER QUICKLY IF YOU LAST HIT WELL. AFTER YOU GET THIS CATALYST REMEMBER THAT BY NOW YOU HAVE SCALED TO ALMOST 60 ap AND HAVE A LOTS OF HEALTH. now if someone trys to kill you you can just stand and tank them. your "Q" cooldown is so small you can spam it on your enemy champ. even if you die you will finish them off just dont use your slowing wall until your dead if you think that you can kill without being killed then pop off that wll first.
After you build catalyst build your level one boots and then finish with a rod of the ages.
the build is very simple from here
1 rod of the ages
2 level 2 boots of magic pen -
3 Archangels staff -- scales of mana and your rod of the ages stacks mana
these items scale so well together its not even funny
4 Void staff or Rabadons deathcap
-either one its totaly preference.
5 Rileys crystal scepter or Zonyas hourglass
---- i always build a rileys scepter over a zonyas.
6 your last item is user choice
--- tanky buld your Zonyas hour glass
--- more ap build another Rabadons ( i take the rabadons )
--- more tank and you can also build a Banshee veil
Masteries are very simple 9-0-21
you need the extra magic penetration and the typical mage masteries are great for karthus
Max out you Q first
-the cooldown on this spell is so short you can easily drain all of your mana within one minute. so be carfull with it and when your ready to attack a champ spam the hell out of it
Take your "E" second
-- this spell is great early game for a gank and late game to escape or CC an entire enemy team while they run away.
Max out you Ultimate second
-- you cannot miss your skill shot on this because if you don't get the kill then you have to wait 3 minutes.
Max out your "E" third
-- This spell scales very well late game due to the scaling off your Ap. it will do almost 350 dps at a full build.
Max your "W" last
if your trying to get first blood ( which i do then you take you "Q" then your "W" so you can slow, flash, ignite, and then q,q,q,q,q,q,q and kill.)
---or you take Q then E if your playing more of a farm first kill second kinda game
All i can say about Karthus spells in a few sentences
1) if you get a Champ with you wall make sure you get your q off at least once or twice
2) His defile does a tone of DPS so make sure you have enough mana left over top at least "e" for a few seconds and q,q,q,q,q,q,q at the same.
3) A karthus ult has no comparison. if used correctly you can achieve at least 10 kills and 10 assists without even getting into a fight with anyone and easily carry the team to victory.
In order to summarize this build i would say that in order to play karthus you need patience.
if youre going to gank you need flash and ignite or flash and exhaust.
--- your "E" is your friend and you can pop it off to ward off enemy champs and to disperse team fights. No one want to be around a karthus popping off bombs and AOE attacks.
You must be very efficient with your mana use early game becaus if you over extend mid and have little or no mana your not ganna escape a gank attemp.
Late game you can use your Ultimate attack when a team fight starts as apposed to using it after the fight is over can get you 5 assists very easily and help your team out greatly because you just dealth 900 damage to each champion!!!!!!!!!!!!!!!!!!