After gaining a level-up or takedown your abilities will have their mana costs refunded for a brief time.
If you're able to slow the enemy with [W] first, your Comet is almost guaranteed to hit. Karthus is also able to quickly bring Comet back off cooldown by landing multiple [Q]s in quick succession.
WhyThe Ultimate Hat
Karthus' [R] has a massive cooldown so every point of CDR is that much more valuable. At max stacks your ult will be up 24s sooner.
Absolute Focus is a rune that can provide incredible amount of AP in situations where you're able to sit back and provide damage. In the case of Karthus, stay healthy in lane and look for an early cross-map [R].
There's not much to say here expect for 'Scaling'. Karth can be one of the hardest scaling AP Mages just by the nature of his [R]. A 40min game means you'll be hitting like a truck.
WhyCoup De Grace
Karth can make great use of Coup De Grace, synergizing well with both is [R] for executions, and when he's in his passive form looking to finish a target.
WhyPresence of Mind
Karthus can actually make pretty surprisingly great use of Presence of Mind with his [E] during teamfights. Pick up kills/assists and that'll empower you to keep your [E] on for free for another 5s.