Across 501 ranked games on patch 16.21 in Master and above, Karthus mains win 58.1% of their games in the bot lane.
Their most common build appears in 82.8% of those games and wins 57.1% of its 415 games.
It starts 
Doran's Ring and 
two Health Potions, completes 
Blackfire Torch, 
Shadowflame, 
Stormsurge, 
Liandry's Torment, 
Void Staff and 
Rabadon's Deathcap in that order, and takes 
Sorcerer's Shoes.
One other build is played often enough in the bot lane to be measured on its own.
In the bot lane, Karthus mains fare best against 
Varus (won 90.0% of 10 games), 
Viktor (won 80.0% of 10 games) and 
Yunara (won 56.0% of 25 games).
Their hardest matchups, the champions that counter Karthus, are 
Jinx (won 42.9% of 14 games), 
Jhin (won 51.7% of 29 games) and 
Aphelios (won 46.7% of 15 games).