check out karthus mains subreddit, it has everything you ever need to know about karthus builds etc.
Personally, I have a few different ways I build karthus, depending on the matchups and the lane im in:
Mid/Top: dark seal + refillable start > Tear, Morello's, Sorc Boots, Abyssal. Then if I'm ahead with some kills, I might upgrade the dark seal to a Mejai's, then start building Liandry's and Void Staff. Then I'll finish off with some AP like DCap or Luden's.
If I'm behind or I'm getting bursted / caught before I can get some spells and ult off, then after the Abyssal I'll build some defensive items. Personally I hate getting zhonyas, I don't feel the few seconds of invulnerability do much for an immobile mage like Karthus. Personally I rather get some pure armor/hp items such as Hextech GLP-800 and Dead Man's Plate. If the enemy team is really sticky/fast and catching me out often, I would considering opting for a Rylai's... but I also feel Rylai's isn't the greatest anymore on Karth even with the slow. It really depends if you want to be more of a sustain / frontline mage or a backline burster with your Q , W and Ult.