[Champion] Dr. Mundo

Pre-game setup
Runes:
Armor pen reds
Scaling health or flat armor yellows
Magic resist blues
Any of the above for quints

Summoner spells:
Ghost is mandatory. It's much, much better than Flash on someone with no hard CC and no way of initiating other than running right at someone.
Barrier is my preference for slot 2, and gives Mundo much-needed time for his ult to heal him.

Other options:
Heal can be substituted as a more team-oriented summoner, but Mundo often is in the middle of the enemy team.
Exhaust is good as a more aggressive option, and is better than Barrier in 1v1 situations if well-timed.
Flash makes Mundo very hard to catch, and Distortion boots give ghost and flash an absurdly low cooldown. Suggested for solo pushing.


Masteries:
http://tinyurl.com/pbz6vcy

Mundo gets much more benefit out of the tank stats than anything else. None of the low-level utility stuff is really worth getting, but the offense tree has some goodies in it. 9/21/0 is alright. 21/9/0 works too if you just want to smash faces.
The really important defensive masteries, in my opinion, are:
Block/Unyielding: Flat damage reduction is invaluable early game. Block will reduce multi-duration ticks (*teemo*) and Unyielding will stop annoying ranged harass (*kennenteemo*). Of course, you still won't win... but you'll lose much more comfortably.
Relentless: Mundo hates being slowed, and this is the only thing in the game that reduces the effectiveness of Ashe or red buff permaslow, Singed and Kog puddles, and other repeatedly reapplied slows.
Tenacious: CC is the worst enemy of melee champs, and Mundo has no way to deal with it other than massive tenacity.
Reinforced Armor: Crits are the main source of damage for physical attackers, and often the enemy ranged DPS will be the only person who can hurt you. For one point, this is a 10% reduction in most damage late game.

Build Order

On load screen you should be looking at the enemy team and figuring out what resists you need, and when. Does your lane opponent do only physical damage? An even phys/magic split? Mostly magic? How about their jungler? Against a normal team in the midgame you should be aiming for a balanced spread slightly slanted towards magic resist, and lategame you should prioritize armor.

Starting items:
Cloth+ward+pots is the default start.
Doran's Shield + pot is for dick lanes. Against ranged champs that you cannot beat (Kennen, Teemo) this is what you should run. The lack of a ward isn't a problem, as you will mostly be farming under tower.
Boots+ward+pots is for lanes that a cloth armor is mostly useless against, such as Brand or Cass.


Aim for Ninja Tabi + Kindlegem on your first back. Against magic lanes go for Boots + Spectre's Cowl. You want to finish Tabi + Spirit Visage as soon as possible.

I recommend these items next:
Randuin's Omen - Try to get this about when the enemy carry finishes IE+PD, or ASAP if they go BotRK. This is the best item in the game for tanking heavy autoattack DPS.
Zephyr - This is THE DPS item for Mundo. The movespeed is incredible, the tenacity is vital, and attackspeed is mundo's main DPS stat.
Aegis of the Legion - If no one else has it yet, this is a solid pick.

Situational:
Warmog's Armor - If you have Randuin's + Spirit Visage and just want more tankiness, get this.
Banshee's Veil - If you need more magic resist, this is your best choice. Also good for facetanking Ashe arrows and the like.
Atma's Impaler - Only as a 6th item when you have Zephyr, but a solid choice then.
Sword of the Divine - It's hilarious, but in general it's too hard to get next to squishy dudes without Zephyr. It has its place, though.
Phantom Dancer - Better than Zephyr if you already have a tenacity item or if the enemy team is CC-light.

Lane matchups
I don't often get the chance to play top, so take these with a grain of salt.
Good lanes (you can farm comfortably while poking, and you can survive all-ins):
Kayle
Karthus
Volibear
Shyvana
Garen

Bad lanes (you will not win, but you can poke/farm with cleavers):
Maokai
Gangplank
Ryze
Cassiopeia
Singed
Darius

Awful lanes (you will either be zoned or farming under tower):
Kennen
Teemo
Swain
Elise

Playstyle notes
Mundo is great at split pushing and getting away, and rather weak in teamfights until mid to late game (when he has two or three tanky items at at least level 2 ult). Early on, focus on creating a diversion for your team by pushing top. Mundo also excels in poke wars, thanks to Infected Cleaver and his natural regeneration. If you must teamfight, draw it out as long as possible. Try to force the enemy team to fight you in many small jungle skirmishes, or set siege to their tower. Mundo also does a lot of damage to Dragon and Baron while tanking them admirably, so force objectives whenever possible.

In full-on teamfights, Mundo can either peel for his carries or dive the enemy. Diving is risky, as even full-build Mundo will drop to heavy focus, so make sure you have someone backing you up. Mundo's lack of hard CC is troublesome for peeling, but depending on the enemy, Cleaver can suffice. If Mundo survives the initial wave of burst and the enemy carry is down, he can often 1v3, or at least kite long enough that his ult comes back up.

Champs that threaten Mundo are: high-DPS carries (including caster-carries like Cassiopeia, Ryze, and Vladimir), fed burst casters (or just two bursty mages will usually do the job), and anyone with healing reduction, including mages with Morello's.

Ticks and Trips
Mundo is just about as straightforward as it gets. Here are some small things that could improve your game.

Activate Burning Agony right before you get CC'd for minimal health drain.

When trading in lane, only fight for the duration of Masochism. Use its downtime to back off or chase.

Mundo has a fairly slow attackspeed, but a quick attack animation. You have plenty of time to move in between attacks; use it.

With maximum CDR, Cleaver's slow duration is slightly less than its cooldown time. Be prepared for the enemy to suddenly speed up right before you throw your follow-up.

Cleaver has a minimal health cost if it hits, but a hefty one if it misses. When possible, angle it so that it at least hits a creep.

Cleaver's damage gets lower and lower as the enemy's health gets lower and lower. Especially with low levels of cleaver, it becomes not worth harassing with it when the enemy is around 30% health.

Timing ult activation is hard - you want to hit it when you're as low as possible to minimize the cost, but not so low that you die before healing. This depends so heavily on every aspect of the game that there really isn't any general advice to give - but it's better to ult early than be late.

+ Show Spoiler [Skill order and pre-game setup] +

Jungle routes, item builds, and ganking
Mundo's speed and damage make him extremely flexible in the jungle. He clears camps quickly enough that he can counterjungle most slower junglers. At the same time, he often gets quite low, allowing a patient or coordinated team to counterjungle him.
Here are some common routes:
Wolves blue wraiths golems red gets you to 4 with double buff.
Wolves (blue->mid) wraiths golems red only gets you to 3, but you only need to be level 3 to gank.
Golems red lets you gank at level 2. If you're going to do this, I recommend starting QE.
Enemy golems enemy red (optionally enemy wraiths) is a good start if you plan to counterjungle heavily, followed by either a gank or wolves (optionally feed blue to your mid) wraiths golems red.

This is the item build I would recommend for most situations:
Machete + 5 pots start
Boots
Ninja Tabi
Spirit Visage
Warmogs
Randuin's

Get Spirit Visage before Ninja Tabi if you're worried about magic damage. Randuin's can be replaced with a bunch of items, depending on your team, the enemy team, and how the game is going, but it's all around a solid pick for almost any situation.
Good items after or instead of Randuin's are Aegis/Bulwark, Zeke's Herald, Phantom Dancer, Sword of the Divine, Ghostblade, and Locket.

Ganks are straightforward: Ghost in if necessary, hit your Q, turn on W, get in melee range, press E and start pummeling them. Counterjungling as Mundo is risky as always, but you chew through camps so fast that it's often worthwhile. If you have a chance to get some lane farm, take it, as you want to hit level 11 (and even more so level 16) as soon as possible.

You can easily take dragon with a bit of assistance at 6, so start looking for opportunities to gank bot around then and carry a pink with you. Baron also goes down pretty quick, so start forcing baron fights early.

In teamfights, poke with cleaver and hang around the sides. Unless you're really fed you can't charge into the middle of the fight and expect to survive. Try to bait spells out and absorb cc for your team; initiating as Mundo is mostly about posturing until you have a good opportunity to engage. Peel for your carry or dive theirs, as the situation warrants.

That's all. Mundo's pretty straightforward, and only depends on hitting cleavers and knowing how to jungle.

The following is a guide to how to play him on Summoner's Rift, since no one plays anything else.
+ Show Spoiler [Skill order and pre-game setup] +

Leveling Mundo's skills is a tricky matter.

Q is Mundo's only form of cc and ranged damage, wrapped into one, that also happens to have built in scaling.
Leveling it up increases the damage and the health cost, but not the slow.

W reduces CC and does respectable damage in a small AoE around Mundo. Good for farming, chasing and baiting cc.
Leveling it up increases the damage, cc reduction, and health cost.

E is an amazing steroid with almost 100% uptime and a low health cost. This is what makes Mundo scary early and late.
Leveling it up increases the damage and health cost.

R is a movespeed boost and a heal over time that costs a percent of your CURRENT health to use.
Leveling it up increases the heal and speed boost.

Obviously you want R at every opportunity.
In lane, 99% of the time you want Q level 1.
From there, you can max Q (the strongest choice, in my opinion).
Q gives you brush control, stronger poke, more dragon/baron threat and better last hitting if you're being zoned. The health cost goes up, but not hugely (30-110, 15-55 if you hit every cleaver), so I often stop after 3 points and start maxing W.
W first is a very specific thing. The previous problem with it was the huge health cost - 20 per second up to 40 at max level. Now that it's much lower, W can give you an edge in pushing and trades against enemy melees.
E first was previously best for trades, but now (with everyone having armor runes and stupid fighting power) one point at level 2 and then Q>W>E seems best.
All Mundo's skills are solid, however, so get what feels best to you and don't sweat it.

Mundo ward his own lane a lot, or else he forget! Has happened before!
Where does Mundo go?
Mundo is strongest solo top, where he can chase people down if necessary and has in general the most favorable matchups.
Mundo solo mid has its merits - no one will see it coming and it is virtually impossible to gank Mundo. This is a solid defensive choice. Mundo also can help out in Dragon fights, and Cleaver hurts Dragon a lot.
Mundo can work in a duo lane, but there are several problems - he eats a lot of damage going in or coming out, cleaver poke has a hard time keeping up with support healing, and Mundo desperately needs levels to stay relevant in teamfights.
Mundo jungle is probably the best place now. With vamp scepter he can stay at near full life maxing E or clear stupidly fast maxing W. He has strong buff control, fast small camp clearing, and is a large dragon threat after 6.
Of course, you are playing Mundo, so go where you please.

Solo lane Mundo
Threaten people early. Cleaver is the second best brush control skill in the game after Orianna's BALLS. Camp brushes and wait for a chance to cleave them a new one.Against most champions, getting next to them at level 2 is equivalent to forcing them back. Feel free to blow your summoners, but be ready for them to flash away because flash is imbalanced. If you force a flash, try to get someone to gank your lane.
If they go back, you go back. Mundo needs health to maintain his pressure.

Most nukers will quickly start outdamaging you around level 4 or 5. Hang back, farm with cleaver, and wait for a gank or level 6. From level 6 on, farm. If you can force the enemy back, go for it and heal up with ult. Keep farming until level 11, or even 16. You are virtually ungankable if your ultimate is up (and you aren't at 100 health or something).

Items
Mundo, like any tanky DPS, has to balance offensive and defensive items.
If you have a strong early game, do not hesitate to build Force of Atmog's (Warmog's Armor, Atma's Impaler, and Force of Nature in that order). Otherwise, grab a Spirit Visage to help you stay in lane longer, followed by Zeal or Recurve Bow. Build towards Randuin's if you need armor. Spirit Visage and MR runes should cover you in the magic department; if not, grab a Negatron Cloak and, if appropriate, turn it into Quicksilver Sash. If you want more damage, pick up Brutalizer and turn it into Youmuu's Ghostblade.
Don't buy Berserker's Greaves. Ever. If their team is CC-light or full of autoattackers, get Ninja Tabi; otherwise get Mercury Treads.
An ideal late-game Mundo will have something like Warmog's Armor, Atma's Impaler, Force of Nature, Mercury Treads, Phantom Dancer and Infinity Edge/Youmuu's Ghostblade.
More likely is something like Spirit Visage, Randuin's Omen, Youmuu's Ghostblade, Mercury Treads, Zeal. Start chugging red and green pots if you think the game will be over soon.

+ Show Spoiler [A list of items to consider buying] +