Make sure you're never in his kill radius: always assume that you have that much less hp whenever you go into a fight.
If you're an ap character, zhonya's hourglass will prevent the damage. If you're an ad character, getting hexdrinker will give you a shield before you die and can be very effective against karthus ult. If you have any way to heal yourself (spell vamp / lifesteal / abilities that heal or heal off dmg), try to attack some creeps while the ulti is going off. If you're in range of Karthus and you have a stun, use it. If Karthus is oom in a teamfight, don't focus him. He'll only be able to ulti when he dies.
If you want to counter karthus, play Soraka and just use your ult right before his goes off to negate the damage. If you don't have a Soraka on your team though, consider picking up a Red Elixir right before a major teamfight. The bonus health it gives cannot be reduced by Ignite/healing debuffs and can often save you from Karthus Ult.