So earlier today I reached level 18 as Karthus (just him today btw) with a full build. I had Sorcerer's shoes, Rabadon's Deathcap, Archangel's staff, Rylai's crystal scepter, and the Void Staff. However I couldn't decide what I wanted last. As the title asks, which would you recommend and why?
Rod should be your first buy. I buy tear, then get rod, then upgrade tear. Doing this build makes you really really weak early because you're not building any damage for awhile, but if you just farm and stall then you're late game is unbeatable.
If it's the LAST item then the best choice is Zhonya's for more damage and the stasis. You have to get RoA way earlier in the game to charge it up.
I play karthus all the time here is my personal build for him
Summoner spells
What i personaly take every time, i like that im able to tp back to lane quickly if i need to early game and then split push and provide pressure with my R for a team fight and then tp in if im needed.
i will take this on arams but i dont usually take this ive seen others take it and it works wonders for them but just not my style.
x2Core Items
or
or both 
Situational items
if you picked both for rod of ages and archangels ignore this
or 
or 
or 
The only reason i like ludens echo is honestly its a bonus 120 ap and its just more ap to stack for the 1000ap dream
I think rylais is probably the best gold value wise on karthus beacuse its only 20 ap less than ludens and gives another 400 health so your a bit healthier On top of all that you get a slow so if you can get a Q off single target you get the full slowing effect (otherwise you only get the 15% slow still good though) if you use your wall of pain(W) to slow to start then get a few Q's up with them you can turn on defile (E) to keep them slowed while you hammer away with your Q.
only get this if your goal is to only survive the whole game ( prolly would pick this up first if your gonna get it) if you can get it to 20 stacks and get the 15%cdr it will be fantastic for your ult being up even more often.
Id only get this if they are like all AP or magic damage champs (like a corki adc an ap support ap mid and an ap top) it gives you some MR which is nice and it reduces there MR so you can hit them a bit more probbably if you doing like a sucide karthus it would be more helpful dont know if you get this effect in his passive though
I used to play him a lot. I would build Tear and Sorc Shoes early.
Final build now would be
Other options are and , but I would build Rod shortly after Tear.
I wonder how 
and would do on him...
Anyway, you want Zhonya's pretty early because you want to flash into the middle of team fights with your AOE on, activate Zhonya's, then start popping your Q as they start to run away, then your R. You want to be right in the middle of team fights.
Also, if you want to troll (and have fun) build spell vamp with full AP build. Super fun.
RoA is best as an Early item on Karthus, so if it was your last slot already, get Zhonya's
Ideally though, you do want to have RoA in your build as a second item after picking up your TotG early. The real choice should have been between Rylai's and Zhonya's, since Rylai's is a much more situational item than RoA. in that case, I would say get Zhonya's if you are getting assassinated, and Rylai's if you want to keep everyone slowed within your E for as long as possible.
