Karthus will be looking to farm as much as possible so he can get his Ultimate R as quickly as he can. You can delay his level 6 power spike by invading his jungle and stealing away his camps.
Once Karthus has his Ultimate R, he can impact the map at any given time. When fighting, always consider that he can use his Ultimate R to turn the exchange around.
Taking away Karthus’ blue buff will make it harder for him to clear the jungle as he needs as much mana as he can get to clear it quickly. If you know his buffs about to spawn and he’s on the other side of the map, try to steal it away.
If you see Karthus using his Ultimate R at the start of the fight, try to CC him so it cancels his Ultimate R.
If you kill Karthus at the start of the fight, try to reposition and get away from his Passive. This will reduce his damage output and make surviving the team fight easier.
If you spot Karthus in a side lane split pushing, you could start a team fight. But, you need to keep in mind that he can influence the existing fight with his Ultimate R.
Once Karthus has his Ultimate R, he can impact the map regardless of his positioning. Whenever you’re fighting the enemy, always take into consideration that he may use his Ultimate R to finish you off.
The more AP and the more magic penetration Karthus gets, the better and stronger he will be. Try to shut him down or constantly invade him to reduce his gold income.
Once Karthus has completed his jungling item, his kill pressure and ability to gank will increase. This item will give him a lot of AP and the first bit of cooldown reduction he needs.