Mordekaiser is rather simple when it comes to gameplay. Play it safe, and if you've got the room to trade (jungle is bot or your jungler is coming), exhaust, pop Predator, and attack. If you've died more than once, don't engage without assistance from a teammate. CS as much as possible and bully them.

Most importantly, have fun!

If you've got any suggestions as to my build, feel free to suggest them - this is my first.

Predator allows you to close the gap, and with exhaust, you gain the two things Mordekaiser is held back by - Mobility and CC.
Cheap shot adds a little damage with your exhaust, eyeball is good for bonus damage, and Relentless Hunter is a good way to gain a little more mobility and not be trapped.

The Ultimate Hat lets you pop your ult more frequently to destroy tanks and use their ADCs against them. It's especially helpful during enemy ganks, where you can kill their jungle or laner and turn it around on the remaining enemy(ies).
Celerity will add quite a bit of AP in tandem with Predator, and with DMP or Righteous Glory it'll be a significant bonus to your first hit damage. I've seen it reach 40 AP.

I'd always suggest starting with a Lich Bane, due to its immense helpfulness with your Q. A proper combo will see three procs with your three Q hits and a ridiculous amount of damage for any level, ranging from 150-1200 damage dealt with LB alone.

Depending on the state of your laning, you could choose between a tankier item (Abyssal Mask versus AP enemies, DMP or Righteous versus AD) and a higher damage item such as Gunblade. Gunblade will double your sustain in lane, its heal is protected by your shield and you can be extremely difficult to kill when using it. Protobelt is another good item due to its gap closing assistance and damage, while it makes you slightly tankier and has CDR to boot.

Speaking of boots, toss a pair in whenever you've got the cash. They're essential for your gap closer and extra movespeed is always useful. Mercury Treads will help protect you from any AP opponents you might face while allowing you to stay closer, longer. Sorceror's Shoes will help you get past a laner who rushes some MR to keep you oppressed. If you're having issues closing the gap without exhaust, switch for a pair of Ionian Boots, which should add a bit of balanced CC/Mobility with flash and exhaust.

Liandry's is vital. Your ult lasts 10 seconds, its burn lasts 3 seconds per ability - that makes it out to an extra 13% of their health, up to 26% if their mobility is impaired the whole time. You might want a Rylai's to pair with it for that effect.

Late game, you're gonna want as much AP as possible for crunching through teams. Rabadon's, Void Staff, and Twin Shadows are all good choices.

The last item of note is the Zz'Rot Portal. If your team is losing in terms of towers, take Herald and Drake, place a portal in lane, and push as far as you can go. The amount of pressure it places on the enemy team will demand they pay attention to you or they'll lose all their towers.

Don't be afraid to break the build and go for something else. Be thoughtful about your options, because they can make or break a game.

Always start W for AOE damage and heals. It'll help you stay in lane for a long time and farm up. Max it first.

Depending on your CS, get Q or E next. Your Q will help a load with last hitting, while your E will help you keep your shield up and has just enough range for moderate poking. Always max Q second, and E third.

Of course, always level your ult up when available.

Your summoner spells should be rather obvious; flash is a no-brainer for its balanced offensive, defensive, and tactical uses, and exhaust will pick up the slack in Mordekaiser's Mordekit.

Mordekaiser has four notable skills that place him above most top laners.

Firstly, his shield and sustain combo from his ult and damage allows him to outstay almost anyone.
Secondly, his ult can change 2v1s into an advantage, and 3v1s into a fair fight. Precautions must be taken when ganking him or you can very quickly find yourself being used to kill your own allies.
Thirdly, his teamfighting ability is amazing - with Q, he oneshots squishies, with W, he can heal himself and an ally while dealing damage and shielding, his E is rather unspecialized, and his ultimate can make a weak link in the opposing team fatal. Very quickly a team can find their situation hopeless, with a 4v6 battle a losing one.
Finally, he can push like no other. Yorick is known as the god of pushing but given a setup of a dragon and Rift Herald, you can annihilate any objective in your way.

Despite his bugs and obvious flaws, Mordekaiser has a lot going for him. Nothing is too terrible about him to be fixed using summoner spells and items, and his late game may be the strongest in the game apart from infinite stackers like Veigar. His damage is insane, his presence is terrifying, and he can support a team merely by being there. Your first games may be rough but sticking through it will get you far.