This is how I build Dr. Mundo. Don't rant or rage if you haven't tried it yet. I feel like with the right build and runes Mundo can be the best tank in the game and thats why we have this build. By the way just to point out but

THIS IS A TANK BUILD

For summoner spells I take Ignite and Ghost. Ghost because of Mundos superior chasing ability and ignite to finish off enemys that your chasing. However their are other spells that are viable.
Optional Spells:
Flash:
This is an overall great spell and can replace ghost however I feel Ghost will help you more with your chasing ability
Exhaust:
This is a good spell but Mundo already has a good chasing ability so its kinda unneeded.
Cleanse:
You probably shouldn't get this spell since Mundo already has escape strategy's(Sadism, Infected Cleaver for the slow and Ghost if you took it)It's a pretty decent spell. Getting out of a stun can save your life but this is more for the heroes who don't have escapes.
Teleport:
The ultimate pusher spell. You can get it but I feel like Ghost and Ignite are better spells.
NEVER GET
Heal: Your a tank, you already have sustainability. Anyways this spell is useless late game.
Fortify:Eh. Not a big fan of this spell. Kinda useless imao.
Clairvoyance: Leave this to the supports.
Smite:If you considered this go find a jungling guide.
Rally:No
Revive:No
Clarity:No

For early game with Mundo you have to poke and farm with your cleaver due to it being spammable. In my opinion Mundo can go top, bottom or mid but I suggest top or bottom. Once you have over 975 gold. Go back to your base(unless if a turret or teammate is in danger)and purchase a Ruby Crystal and Boots of Speed. Overall the key to early game is farming and harassing with Burning Agony and Infected Cleaver.

By this time you should have 3 items on the list(not including Dorians Shield)and have over 1000 health. Since you are the tank you will lead the battles while the carrys do the damage behind you. If you are being chased I suggest using your ultimate to get away due to it's speed boost and the health regen. When chasing enemys hit them with your cleaver and run up with with your W. If they get out of your reach again then Ghost up to them and finish the job.

You should be working on Randuins Omen by now and able to tower dive with out taking that much damage. You should still and should always be spamming your Infected Cleaver. If you get to their turrets your Mascochism can help with melting it down. Sell your Dorians Shield and buy a Guardian Angel after your done with Randuins Omen. If you have enough time, you have the option to buy a Thornmail if you want. Once you buy your Guardian Angel you will be unstoppable in the battle field and you'll see that health bar on the Nexus go down faster then a bullet.

This is one of Mundos most core spells. You can basiclly use it for anything. Farming, poking, catching the enemy, getting away from your enemy, etc. You can choose to focus harassing or farming early game but I choose to focus farming because this is an expensive build but that doesn't mean that you shouldn't poke at your enemys(unless if its Soraka,Vlad,etc)

I focus health because Mundos a tank obviously but for my Quitensstances I go armor because a tank cannot be a tank without armor. I once killed a 4k health Amumu who had no armor when I was playing Singed with 3k health and armor. Also the health per level is because you will be unstoppable late game.