This guide is not about the most optimal way to jungle Doom. Doom in the jungle is relatively recent as of writing this guide, so there's a lot of experimentation to be done. This is merely a straight-forward way to jungle Doom. It is a foundation.
Get a Ring of Basilius and a single set of Tangos. Head to one of the hard camps. For both teams, this is the camp adjacent to your middle lane's tier two tower. Devour the largest creep in the camp. Use the spell immediately. Don't whittle the creep down first. Finish the rest of the camp. You will probably need to use one of your Tangos. Do so. Now check your gold. Do you have 250 gold? Yes? Send that Stout Shield. No? Well, when you do, get it.
Now head to the small camp. You might need to use Scorched Earth here and/or a Tango. That's fine. A general tip is to always target the creep that deals the most damage (usually the largest) or makes the biggest impact. For example, Satyr Soulstealer deals 24-27 damage, so he should be taken out before the Satyr Tricksters, which deal 7-10 damage, and while the Forest Troll Berserkers do more damage than the High Preist, the latter's heals will slow your clear.
Now you should be ready to go to the hard camp again. Devour should be nearing the end of its cooldown. Go repeat this process, using the last Tango at the hard camp. From here, it's all about using Scorched Earth when you absolutely need it to tank the camps; dealing damage and regenerating health. The passives from the large creeps; be it additional armor, mana regeneration for Scorched Earth, attack speed or additional damage to clear faster, or even health regeneration, should be enough to keep you in the jungle.
But, there's a condition to fighting hard camps. Furbolgs are your worst nightmare. The one that remains after you devour the largest one will hurt you much more than you will hurt him. This is the only hard camp where I recommend you not fight the left-over creep, instead opting to devour one and come back to devour the next one later. Don't just ignore them and go to the other camps. You need to clear them out so that camps that you can clear will be available.
You are going to rotate back and forth between the easy and hard camp, using devour to make the hard camp easy to clear. Do not use creep abilities unless you absolutely have to. Don't feel pressured to gank with them either. Your mana pool is simply too piss-poor to gank with them, and even if you were successful, you would have to go all the way back to base to get that mana and perhaps even health back, delaying your jungle. Doom is not a ganker. He's not an early game hero at all. He's not a tank. He's not a nuker. He's a late-game carry who, with the right items, can do those things, but doesn't do it from the start.
The second you get the money for Gloves of Haste, send them your way. This item is going to make clearing the camps much easier, but you're still going to be rotating between the two. Getting the remaining 1,400 gold to finish Hand of Midas should come quickly. Once you have the recipe, send it your way and let out a sigh of relief, because your jungle is now guaranteed. The hard part is over.
Devour and Midas will allow you to clear two hard/medium camps instead of one when you're done with the easy camp. Use Devour and Midas at two separate camps, unless it's a Furbolg camp. You want to alternate these two abilities so you always have a clear open. Some camps can be cleared easily without either; such as the Wolves or Medium Satyr camp. You can use Scorched Earth to tank others, though. That's all you need to get up to speed. From here, you pretty much snowball in terms of farm. Every camp is open to you. No other jungle items are needed. You can start working toward your core items, whatever they be.
This should take you twenty minutes. You should have your Midas at around 8-9 minutes. At around 20 minutes, you should have a big item like Radiance, perhaps. I recommend Radiance in conjunction with an armor buff from the jungle because Scorched Earth and Radiance really hurt the enemy team this early, if you leave the jungle to help your own. But, more importantly, it accelerates your farming to another level. You can easily make enough gold in the next 5-10 minutes to buy one or two more big game-changing items. That's not counting gold you get from outside the jungle.
I feel these are the best abilities/passives to have. Some of them may cost mana, so use them wisely.
I have tried a few builds. I've tried Stout Shield, Quelling Blade, Tango. I've tried Ring of Protection, Stout Shield, Tango. I've tried Ring of Protection, Quelling Blade, Tango. None of them give me the feeling of security like Basilius and Tango into Stout Shield. Another big item people are considering are Tranquil Boots. Here's my rationale for not recommending them.
First, they delay your Midas and jungle in general. Your Midas lets you clear fast and more often with both the attack speed buff and the active. That means less damage done to you anyway. Your Midas is going to get pushed to the point where you should have a big item like Sacred Relic if you build Tranquil Boots.
Second, the active on Tranquil Boots is atrocious. You're not Axe. You don't clear camps fast enough with Helix that the active makes up for the little dings here and there. You're going to take damage that makes the active trivial. Second, it wastes mana. The one or two times I tried Tranquil Boots, I was constantly out of mana from spamming the passive thinking it was going to make a difference. Are you dangerously low? Just send a salve. It heals much more for a fraction of the cost.
Third, if you are conservative with your mana, and the rotation above should ensure that, then you will always have mana to use Scorched Earth. The health regeneration that provides is more than enough, but if that isn't reassuring enough, what about the passives you get from devour? Virtually every one gives you sustain. There's only one advantage to Tranquil Boots. They're boots. You do need boots, but you can farm Phases easily when you actually need them. But, don't take my word for it. Experiment for yourselves.
I don't level Doom at 6 because I'm not going to be leaving the jungle. Its mana cost is too much for Doom's ****ty mana pool anyway. Devour and Scorched Earth are both needed for the jungle. Level Death has been recently changed to be more useful later game considering the bonus damage is now 20% of the target's maximum health, so take that later.