Well, this is how I play jungle mumu. Is it set it stone? no. nor should any build be, but this is usually what I end up getting. I know you might disagree with my choice of ghost over flash, or plentiful bounty over utility mastery, or avarice quints over fortitude, or rylais over randuins, or GA over BV, or maybe my skill order, but everything is done with a purpose, so please read my reasons, understand how it suits the play style, and give it a try if you are so inclined - i'll tell you right now, if you can land a skill shot, you wont be disappointed.

besides my quints, i think everything else is pretty standard and obvious, so let me talk about the quints a bit. What do gold quints give you... 6 gold a minute? that seems like nothing right? Well at 20 minutes, that 120 gold each (360 gold total). Great you say, 120 gold (or 360 with all 3). Well whats so great about the quint you would have picked otherwise? Besides mp/5 quints, crunch the numbers on any other quints you would put on mumu. Guess what, most of them will have an in-game value of about 120 gold or less. mp/5/lvl quints are about a value of 240 gold - so if you know you want mp/5, get those quints instead, but otherwise, you might as well do avarice quints and then have the flexibility to buy whatever you want with the gold instead of being locked in before the game even starts, and on top of that, the longer the game goes the better the quints are. at 40 min? 240 gold a pop (720 for all 3). At 1 hour? 360 gold (1080 with all 3). That is over 1000 gold you got from those 3 quints if you pull up on the hour mark. You cant beat that, especially when you playing a character that isnt supposed to get a huge farm. So hate if you want, and use different quints if you want, but at least come to terms with the fact that these quints arent that unreasonable and actually offer you some nice flexibility.
Of course if you find yourself having trouble in the jungle, go with flat HP quints, but it's definitely doable without and the 78 HP you get from them is negligible late game.

No real surprises here, just want to note that considering tantrums CD goes down when you get hit, dodge isnt that great of an investment. so you could technically put the two points from evasion in strength of spirit.
Utility mastery v plentiful bounty v greed
Money wise, greed is the winner. I mean it nets you 6 gold a minute even if you are picking your nose the whole time, but frankly, you're a jungler, and u can make better use of the lower CD on smite. So the fact that you might net a little bit less from the skill alone is irrelevant, since you will probably net more overall since you can kill minions faster and control dragon/baron/buffs more effectively with a lower smite CD.
Utility mastery, however, is the winner, especially with how mana hungry mumu is with despair in the jungle. With other junglers you really make better use of the cd on smite, but with mumu, blue buff will speed up your jungle much more than a 5 second cdr on smite.

Start with a standard jungling set up. Whether you get boots or a chain vest next depends on how the map is looking. If you think you can get some help to take down dragon safely, get the chain vest and a health pot. the chain vest will make all the difference in tanking draggy early, but if its not looking good I would grab the boots to move through the jungle quicker and pick up a gank. So in the item sequence I put the chain vest first to draw attention to that option, but it is situational.
I've seen a lot of gold/sec items in different mumu builds, but I dont think it's the best use of item slots for mumu. If you get HoG or PhilStone it will take way too long for it to pay itself off and neither item is super useful to mumu on its own. I realize the same argument could be made about my gold/sec quints, but the difference is those kick in from the first second in the game, and never have to be forfeited for something else mid game, so the longer the game goes, the more valuable those quints become. An HoG, however, is a waste of space if you cant turn it into a Randuins Omen for 27 minutes just because you want it to pay itself off.
Sunfire cape is a great farming tool. it lets you quickly push lanes when coupled with despair and farm very effectively. The synergy between sunfire, despair, and abyssal scepter is also a force to be reckoned with. If you get far enough along to get rylais in there as well, then enemy champs better hope they can kill you before you drain them to death because they are likely not getting away without using a flash.
Aegis is a standard item for mumu and should really be built unless you are absolutely stomping the enemy. Then Id maybe go for a deathcap.

Guardian Angel v. Banshees Veil
I realize these two dont have to be mutually exclusive, but if you want mumu to do something besides be a CC bot, you will need some offensive items. That being said, you are a tank, and although BV is nice for survivability, it is like wearing a huge sign that says "DONT WASTE YOUR STUN ON ME" - when in reality its better that the enemy blows their cc on you than on your carry. So i would rather not have bv (even though it gives some nice health), be a CC magnet and then come back to life if i die. Arguably a GA could been seen as a "dont bother trying to damage me i will come back to life" sign, but if you are in their midst and doing your little AOE damage, they will likely waste their CC on you if they can, which frankly i think is more important. So knowing that you will be a CC magnet, you must have merc treads.

Randuins Omen v. Rylais
Also not mutually exclusive, but a little redundant if you have both. With rylais, every time you despair or tantrum you just did what randuins omen's active does. They both have health, they both give you an AOE slow, but rylais gives you AP which means more damage, gives you a spammable slow, and past 200 armor (which you have from other ****), randuins armor bonus is really a diminished return.

The skill sequence is a little peculiar because I dont max despair until late. I love despair early on because of how great it is in the jungle, but honestly a % dmg spell is much more effective late game, but that is not the only reason I dont max it until late.

Mumus q and e have huge drops in cooldown as you level them, w does not. So while w is very useful, being able to get 2 bandages off in a clash is the difference between killing one or two enemy champions with a buddy.

Tantrum also has some nice cooldown times as it gets leveld, but with the little armor buff mumu got on it, it is especially useful at high levels. The sooner you get that armor, the more it will matter, as 6-10 armor is not that important 30 minutes in.

So the goal is definitely to get skills according to the stage of the game in which they are most effective.

Why ghost over flash? Easy, despair + sunfire + abyssal + rylais. I dont just want to be next to an enemy for a moment, I want to be next to them and stay next to them. Even just running next to an enemy means he is dying.

start blue buff. the problem is, everyone knows you need blue, so dont pick your skill until the last moment. If it looks like blue is going to be too dangerous, take tantrum and start at wraiths. However the skill sequence I have is assuming you can start at blue and go wolves => wraiths => red => golems => gank/invade => 3 minor camps => dragon