Jungle Mundo goes where he pleases!
Open up discussion on whether or not your team is going to do the dragon. Either way, step 1 is the same, but step 2 can change depending on your side and if you are dragoning or not.
Right off the bat, buy your cloth armor and 5x health pots. Immediately head towards mid/dragon and throw cleavers in all the bushes. If you hear a THWACK, that means there's somebody there, now its time to back up to safety, but at least you have alerted your team that there is an early dragon attempt going on.
Step 2 if dragoning)
While you were messing around mid/dragon, your team should be scouting/setting up in the blue jungle around the lizard. Around the 1:20 mark, you head to the lizard yourself and lead in a quick kill, be sure to drop 1 health pot while tanking the lizard and use your smite to finish it off. Welcome any team ganks the other guys might have in store for you as Cleavers are ridiculously powerful at level 1. Nothing should really be much of a surprise if you have checked the jungle thoroughly anyway.
After the lizard is down, head straight for the dragon, drop your rally and another health pot on the dragon and he will be down in about 8 seconds. This is especially effective if you have 1 teammate stay out of the dragon fight as 4 people in close proximity will mean 4x level 2s @ the 2 min mark of the game.
From there you can either pick up the Purple golem buff for one of your casters, or head to bottom/mid for an easy gank depending on how things have played out.
Step 2 no dragon)
If your team doesn't want to do an early dragon kill, get out of the area at around the 1 min mark. Tossing cleavers at their team while they are trying to set up a dragon kill, alerting your team that they are there, and maybe leading up to a counter dragon gank is 1 thing; sticking around too long and getting killed due to a well placed flash/exhaust or something is just dumb. Don't stick around long enough for this to happen.
Head towards your jungles lizard. Check the surrounding brush and be ready to start the fight by standing directly behind where the lizard will spawn. Make sure you are in place by the 1:39 mark.
Cleaver, pot, hit hit hit, Cleaver, Smite, hit till dead (in that order specifically) will give you a 6 second lizard kill and mopping up the small lizards will get you to level 2.
Clear the other two camps in the area starting with the golems/wolves camp, then heading to the wraiths camp should get you to level 3.
********From here on out, you put emphasis on ganking and jungle only when there are no exposed lanes or you need a lizard buff.********
At this point you are hitting for about 120dmg a swing. Crits will hit in the 250-300 range, and cleavers snare people enough that they won't get away without flash.
Start with mid lane. Wait in the brush for them to push too far up. Have your teammate lead with what the other guy will think is a stupid move. He can back out right away, all we need is for the enemy to over commit. Landing a cleaver on the charge will give about a 90% chance of killing them. Landing a cleaver after the charge will give you about a 60% chance of killing them.
If the gank went well and you have 2 in mid with no opposition now - push the tower, drop rally and you will have mid tower down by the 5 min mark. Watch for teleport after the rez or top/bottom to come protect the lane though.
Head back to town, finish your Madred's, and if you did particularly well, pick up a vampiric scepter, then back to the jungle, you could probably use the other lizard buff right about now so go check it out.
You should be level 4 right now, so no single character can kill you alone in the jungle. Charge into their jungle, take the lizard buff, and back to ganking. Top and bottom are easier to gank if they are pushed up to your tower than mid was. You have to watch all 3 lanes and be ready to move in at any moment though.
Continue this way until the teams start to move together. It should happen rather quickly as you will be eating them up while they are spread out in the lanes.
Hopefully, you can finish your Berserker boots and Malady before you get into the team fighting phase.
Jungle Mundo is not a tank, but don't be afraid to initiate fights either. The only things that can stop you are stuns, blinds, and an executioner's calling if they spent all their early game gold on it. Hitting your ult about 5 seconds before a fight actually starts is very beneficial as losing 20% of your HP after the action has started can actually be the cause of death often time. Between your 200dmg Masochism swings, your 1.4 attack speed due to your items, Malady's 15% life steal, and dropping Rally if you really need it, you should have no issue keeping yourself alive through just life steal. 1 Crit can hit for upwards of 500 dmg and heal you for at least 70hp.
***Other tips and tricks***
-Practice your cleaver toss when charging in. Psychology dictates that enemies will take a step towards their base right as you get into cleaver range. Throwing a cleaver 1 step behind the enemy when charging is a great way to greatly increase your chances of landing it. Always throw cleavers where the enemy is going to be, not where they are. Most times people run in a straight line away from Mundo as fast as possible because he will wreck their face (and everything else) if they don't.
-People will position themselves with creeps in between you and them so that you can't hit them with cleavers. If you find they only have 1 creep in the way, smite the creep with an immediate cleaver is a great way to catch them off guard and most likely close the deal on a kill.
-Look at the item build. All emphasis is placed on attack speed and nothing else. Masochism gives you +100dmg and further additional damage based on how much HP you are missing. The lower you get, the faster you kill people. The best way to increase Mundo's damage is through attack speed. If at any time you feel a Elixir of Elusiveness will benefit you more than moving the item build forward, go for it. It a great use of 300g at any point in the game.
-Madred's Razor is gotten early to help with jungle/pushing speed and to help keep your HP higher while not in an actual fight. There is nothing better than critting a creep for 900 dmg at level 4 (healing yourself for 135hp in the process) or clearing a whole wave of enemy creeps in 6 swings (happens quite often between crits and madred procs).
-Once you get to level 8, always go into fights with Burning Adrenaline on. It has a high health/sec cost, but you drain more than that by attacking, and the damage it deals along with the CC-reduction it gives are definitely worth it.
-If the other team is running in a tight group and yours isn't so much, don't be afraid to solo push.
If you see their whole team pushing on bot and you're on top. Do a deep push until they turn to come stop you. You can usually time how long it will take their team to cross the map. Hit Sadism and Burning Agony and hightail it out of there right before they can get their hands on you. If you have Zeal, nobody can catch you without a solid stun or immobilize. Even then, if only 1 or 2 guys catches you, they won't kill you before the stun/immobilize is over and you zip away anyway.
-Most importantly, say your name a lot so you don't forget it!