Always

]Harass, harass, harass. Any time you have Siphon of Destruction up, use it. Do not be afraid of the health cost, it is quite literally unnoticeable.

Tradeing

Stand as close to the enemy minions as possible, they are your friends and they will increase your shield substantially.
Watch the cool-down of your ultimate. If you are near death, use it on the squishiest target possible, even if they are not low. This will give you a large burst of health and very high health regen. Now that you have all your health back, kill whoever you ulted and win the resulting 4v6.

Getting Ganked

If you are getting ganked, do not be afraid to turn and fight. Due to Mordekaiser's unique kit, the more enemies you hit with your attacks, the better. Just ult the one with the lowest health after you take a few hits, focus them down, and use them to finish of their buddy. You may do whatever you please with the clone after the fight, but taking a tower is usually the best option.

SPAM ALL THE THINGS. Use every ability you have on as many enemies as possible, as often as possible. This will give you (A).A huge shield, and (B). All the free health in the world.
Ult the adc, focus the adc, and kill the adc. Use the remaining clone to make the enemies run screaming.
Your ult has a surprising range. Use it to execute fleeing enemies.

If you want to kill Mordekaiser, reduce his healing, build MR, and if you get ulted, run like hell.
Quicksilver Sash will cleanse his ultimate.
If you can stuns and silences are his kryptonite. If he can't use his abilities he will melt like butter that has been thrown in a volcano.

Counterpicks

Garen mid is suitable option when Mordekaiser is involved.
Anivia will eat his soul the entire game. She may not be meta, but she sure as hell can keep Mordekaiser down and out in every fight that ever happens.
Leblanc and Kassadin both counter Mordekaiser for the same reasons: mobility and crowd control. Both can just jump on him and smack him to make him deal no damage and have no escape.