(Formerly Silver IV) Mastery 6 Karthus main, here's some stuff that I know along with some tips that I picked up from a Diamond 4 Karthus main :P
When going mid, it's fine to get a Doran's Ring, but I personally like getting The Dark Seal. It gives extra potion power, and if you get fed, you can build into mejai's as well. (it's not as bad on Karthus as you think, since you can still get stacks while in passive). If you can't build into mejai's however, it's ok. I don't really build boots anymore because I find getting another power item is better. According to the Diamond Player, you should never build more than 1 defensive item if you need to.
My usual Mid Build is
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>
+
>
>
>
OR
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>
>
(if fed, get Mejai's in place of Void Staff or Scepter/Hourglass)
Now there are some situational Items I go to just in case. I usually will replace either Void Staff or probably Abyssal/Zhonyas for one of these if I do:
: If I wanna go my "NasKarthus" build (goes with ghost as well)
: If I want more tank/man and an active damaging slow
: If I want more overall slow (however I know it's gonna get nerfed soon, so I've been staying off this for a while)
and if you want to go Jungle with him (it works suprisingly well in this new season), you can basically just use the same build, only instead of Dark Seal, you build toward Runic Echoes (w/ Blue knife usually) and then I usually sell it to get my void staff.