Infected Cleaver: I take Infected Cleaver at level 1 for the slow and harass-ability; maxing it first for optimal damage.

Burning Agony: I take Burning Agony at level 2 and max it second. This ability is effective for chasing acting similar to a Sunfire Cape. When combined with Warmogs to balance out the health cost and combined with the slow from Infected Cleaver it makes it difficult for anyone to escape you. This is why I recommend Boots of Swiftness for chasing down enemies. This is the only one of Dr Mundo's abilities that has a scaling effect and it rolls on AP, though I don't recommend taking any AP items beside Rylai's Crystal Scepter since focusing health and health regeneration is better for Dr Mundo's survivability and ability to engage and chase.

Masochism: I take Masochism at level 3 and max it last. This ability increases Dr Mundo's AD for a short time, giving bonus damage based on the percentage of health he is missing. I take 1 point in the ability early game to help with pushing and damaging turrets, but max it last because the bonuses depend on missing health - this build with my recommended items is designed to keep you at least half to full health most of the time, therefore making the less useful than Infected Cleaver or Burning Agony.

Sadism: I take Sadism when available starting at level 6. The regaining of health over a period of time is useful for engaging the enemy team, while allowing your teammates to take down the enemies one at a time. It is also good to save the ability for escapes and engaging a group of enemies to help a teammate escape.

For Masteries, I recommend 0-30-0, putting all points into defense and being sure to put 3 points in Mercenary(granting bonus gold on champion kills and assists). Played in the full tank roll with proper use of Burning Agony, Dr Mundo should primarily get assists with the occasional kill from Burning Agony in a team fight or from the Teemo in your lane who thinks he can 1v1 a Tank with a Sunfire Cape passive that has a slow and goes where he pleases.

Another option, but less recommended is 9-21-0, focusing on health and defense with a small amount devoted to AD, attack speed, and armor penetration.

The recommended item build is useful for any game, primarily those with more AD than AP. In the case of an AP heavy team I typically substitute the Sunfire Cape for a Force of Nature, otherwise I just focus armor and health. Other late game items you can substitute in place of additional Warmogs are Rylais Cyrstal Scepter or Frozen Mallet.

Farming is especially important early game for rushing your first Warmogs. Once you have that item you should be able to almost indefinitely hold your lane until you need to go back for items.

Laneing Phase(Teammate): You want to do your best to keep yourself between the enemies in your lane and your lane mate. Take the damage for them where ever possible since your passive will heal you overtime. This will help your teammate stay in lane and allow you to keep constant pressure on the enemy tower.
Laneing Phase(Solo): Personally I usually stay out of the solo lane unless none of the other players on the team can take it; however, Dr Mundo is a good option for soloing. The passive health regeneration will allow you to stay in the lane, but regardless, defensive play is required. If laneing against melee champions, be sure to build Giants Belt first. If laneing against ranged champions such as Caitlyn, Sivir, or Graves, start building your Warmogs with a Regrowth Pendant - you'll be taking more hits so you'll need more health regeneration to hold the lane.

Mid-Game(Teammate): At this point in the game, the laneing phase is ending. At this point you should have your first Warmogs, and Boots of Swiftness. You should have sufficient items to help push any remaining outer-towers. Dr Mundo's slow and natural survivability make him a good Ganker, especially in the mid-lane.

Mid-Game(Solo): By this point in the game you should have at least one point in Sadism. You can allow the enemies in your lane to push you back and bring down your health so they'll get over confident and dive. Once you bait them in, you can cast Sadism to regain health while using your slow to get an easy kill for you and/or your Jungler.

Late Game: At this point the majority of the game should be team fights which as the undying tank, you should be initiating. By running in, the enemy team should focus you, at which point you can use Sadism to escape while disorganizing the enemy team and allowing your teammates to pick up the kills.