Karthus doesn't need a big rework, he just needs a touch up
see title
I'm not really a Karthus expert; I don't have ten thousand games on him or anything, but I like him a lot so his changes are relevant to me. Riot's said that they're hesitant to change him because all of his spells are iconic to who he is; he's a pretty unique champion in how he plays and feels. And that's understandable; glaring weaknesses and meta state outside, he's stood the test of time very well.
But Karthus needs help. He can't contend with the rising state of mobility today, and Riot was very quick to drop the hammer on him when he could have potentially been strong. I don't think he needs a very big change list to bring him up to par; he's a damage machine, and that should be his strong suit - his value to a team comp, being able to out-DPS anyone provided that he lands his Qs, and if he can't, he gets a second chance to wreak havoc with his passive.
I think the biggest change I would suggest would be to his Wall of Pain. Just an idle thought here, but what if it were actually painful? To suit Karthus with an ability to punish mobility without simply applying a Grounded debuff like Cassiopeia, you could change it so that after passing through the Wall, you take damage for moving away from Karthus, bonus damage if using a dash or blink (excludes Flash.) This way, his wall punishes mobile champions more, but doesn't completely disable the option to use your dash to get away. Do I want to wait out the slow and just move to dodge his Qs, or can I risk taking the damage and dash away from Karthus? I understand that this would probably require power taken out of his W elsewhere, but I think it's a decent opening for discussion.
Aside from that, really the only thing I would really like is his Q to be more consistent. I don't think it's just me, because I've seen people comment that his Q feels inconsistent with its delays. Correcting this to what it's supposed to be would, on its own, help Karthus greatly. I don't think his E and R need major changes; they do their jobs fairly well, and Karthus can more comfortably fit CDR in his build for more ultimate use with the coming item update.