Bramble: If your enemy is a healing champ (Irelia, Aatrox, Nasus) you have to rush this item.
Ninja Tabis: If your enemy does most of his damage through Auto-Attacks buy this (after Bramble if he/she also heals) (Irelia, Nasus, Jax). (If your opponent likes to run away (GP) you can also buy it)
Mercury Boots: If you opponent is AP you most likely want to rush it (Teemo, Kennen). You will need the movement speed and MR.

Possible Items:
Thornmail: If your opponent/the enemy team has Lifesteal
Gargoyle: Very good if you need AD and AP. The shield is really strong (700-1.1k Health).
Rylais: Good end item for some better chase and damage
Dead Man's Plate: If the enemy team has alot of AD but no Lifesteal.
Spirit Visage: If the enemy has alot of AP. Nice bonus to your shield.
Morello: Against a strong AP Healer (example: fed Gwen with Riftmaker)
Zhonyas: When the enemy team has a very hard carry noone can beat. Ult him to take him out of team fights and then Zhonyas so he won't kill you. Only do it against opponents that fed, that even if you build against him/her you can't beat.

I created this page for myself, put everything what i learned as a 1M Morde Main here to find ways to improve. I will also add some stuff from Master+ Tutorials here so i know what i need to improve/to do.
If you also want to learn from my experience and summaries from Tutorials... here you go.

Mordekaiser is weak before level 3 and will loose almost all 1v1. If possible, AA or Q Minions. If possible also the opponent.
BUT in most of the cases just don't risk to get full-engaged (exceptions: low life enemys, weak early enemys like nasus). If you get full-engaged by someone you can't defeat use your E to draw him back if possible, else Flash.
If you get full-engaged, you will often loose a huge chunk of your health which you will need to regenerate over time with you W.

From Level 3 you will start to take trades or even all-ins. This is often just a skill-based fight. So you need to know your matchup and how to win the fights.
Examples where you can loose:
-Getting engaged by an enemy when you are low life
-Getting engaged by an Irelia who has full stacks
-Getting engaged but missing your Q and E, therefore not getting any Passive Stacks
on your opponent.
-Igniting AFTER Darius healed himself up with his Q
-Using your shield at only 50% full
...

If possible you want to Engage an opponent with your E+Q giving you already 2 Stacks. In pretty much all cases you win the fight if the opponent does not run away. And if he runs away you dealt some good damage to him.
If you have your W on cooldown you most likely don't want any fights to happen.

Sometimes it is possible that you do mistakes and are behind. When that happens it is safer to stay near your tower and wait for your jungler and get all the farm you can get. Just don't do anything where you get caught by your opponent.

The best location where the minions should be is right in front of your tower (except Level 1+2, there you should let them in tower and take only safe gold). That location has 4 advantages:
1. The enemy jungler can't gank
2. Your jungler can gank
3. If you loose the fight you can escape
4. If the enemy looses the fight he can't escape that easily

There are 2 ways to achieve that:
-to freeze the Minions in front of your tower, your opponent should have 2-3 Minions more then you. If the enemy starts killing minions, you will kill the same amount.
-if the minions are frozen near the enemy tower (you have 2+ more minions there), kill all enemy minions so the wave gets destroyed at the enemy tower.

Be careful if your opponent reaches Level 6 or can reach it very soon. He will most likely immediatly jump at you before you reach Level 6 yourself and you will loose in almost all of the cases.

When to Ult:
-You can also abuse the fact if you reach Level 6 before the opponent to get a safe kill.
-Only ult if you are pretty sure you can kill your opponent.
-To escape ganks/1v2 them

The safest way to be safe of ganks is to freeze the lane near the tower. But this is not always possible.
These following factors determine if you die when ganked:
-Who the enemy Jungler is (You can't always see him, even when warded)
-How well you warded
-If you have seen the enemy Jungler on the Map before
-How far out in the lane are you?
-Does your Laner has some CC to stop you when trying to run away?
-Are you Level 6 and can 1v2?

It is hardly possible to never die through a gank. Sometimes you just have to take the risk of being able to get ganked to bully/kill your laner. I mean... else why would the Jungler even exist?

In almost all cases you want to Ult the enemy who is the biggest problem to your Team. It can be an Amumu before he Ults, a fed ADC...
Exception: When fighting an Objective you most likely want to Ult the enemy Jungler so he can't Smite and you Steal.
Exception 2: You are caught in a Rumble Ult etc. or surrounded by enemys who try to kill you... Ulting can be a good way out

(No specific order)
1. Getting caught/engaged before level 3
2. Getting caught/engaged in a fight which you can't win (Stronger enemy, low life, in enemy tower where you can't fight back)
3. Doing a 1v1 mistake (Miss Q-E and get engaged...)
4. Enemy having Ult/getting Level 6 when you don't have Level 6/Ult
5. Getting ganked and die
6. Overstaying when you are low life
7. engaging/getting engaged without abilites

8. Ulting the wrong opponent in Team fights
9. Getting caught in mid game.
10. Not being there when your team fights for objectives and the whole team dies.

1. Farming (you want an above average farm, 7+ cs/min early, 5+ cs/min late if possible)
2. Warding
3. Grouping when necessary/Helping your jungler when near you in need