I usually take ghost with ignite on karthus.

ghost is underrated summoner spell and in team fight u can position your self better, before dying (u probebly will be the first one to dy?!)

flsh is good on every champ except singed.

ignite is good for early game power and first blood.

exhust is pretty good so u can land q and burst the opponent.

teleport is always good after lvl 3 when the top or bot commit to a teamfight and u surprise them, force disengage, and split push.

if u r laning against ad assassin like zed rush seekers armguard. if u r ahead of your apponent get deathcap after roa(roa is usually the first item that karthus completes). if mid game the teams r dead even or your team is behind rush zhonya's and Seraph's Embrace for extra damage and durability.

21 point on offensive mysteries is a must on Karthus. Utility and defense is optional , depending on your style.

this is my personal favorite,
i call it vlad runes(the one showing).
If u r against ap or ad assassins be sure to have armor yellows, or blue mr.
u can use flat ap but by lvl 7 u can fell the more boost on your ulti

after the jangler and top laner engage, don't be shy to rambo in, besure that after your death your teammate bait the fight to continue on your deadbody. after u died u have small window (around 3 seconds) to cast more q before u ult (i ult right away i die because i dont wanne risk my ult for more q). Overall karthus aoe damge and passive and ult is game changing and makes him easy to be used in teeamfight. besure to be on the same page with your team and commit to a fight when the enemy is using their ulti