Welcome hopefully youll find my build good for conquering the enemy team so they cry and ragequit first ill explain why i picked my summoner spells first clarity because while karthus so should always be mid cause he takes priority over every else but clarity will help you get your mana back while your spamming Q to annoy and slowly peel away there health. Second spell is because when your chasing your kill if your ulti is on cd or your saving it for teamfights and or support

My Item build goes first Sapphire Crystal because Karthus needs mana early game/mid game. for mid your gonna wanna get a health and mana potion cause you tend to try to tower dive at low levels cause you wanna get the kill. Tear of goddess because you will be spamming Q to the max.
Archangels because you know its pretty much tear of goddess upgraded. Mejai's soulstealer because you will tend to get assists/kills because of your ultimate if it isnt working for you please sell it and buy frozen heart.sorcerers shoes cause of the magic penetration or you could go for Ionian boots of lucidity for the cooldown reduction.Rabadons cap because as karthus you wanna pack mana and ability power.Zhonya's hourglass because of the extra armor/ability and the ability power.last is void staff because of the magic penetration and the ability power

My runes Are pretty much Magic Penetration/Abilty power/Mana regen Because early game karthus is a mana burner. Same for quintessences Ability power/Cooldown reduction/magic pene

While Karthus is a really good farmer at late and early game you might wanna take a detour sometimes to go to the jungle because it offers you golem buff that will help you alot and money which karthus will benefit from.