I am B1shop. I am a Silver Braum main who has been logging a lot of time on Braum lately. And while I originally thought of him exclusively as a support, I always thought his damage output was high enough to try him in a different position. So, one day, I tried him in the jungle. Ten kills later, I figured out that he wasn't such a bad carry.
The trick to Braum is in his Passive, Concussive Blows. It helps in duels, ganks, clears, teamfights, you name it. And his passive is precisely the reason that you build Devourer on Braum rather than Warrior (and don't get me started on that disgusting tank one). With Devourer, With two Auto Attacks and a Q, an enemy is now stunned, which is very fast, given the fact that Braum also gets additional magic damage from his passive and from his Devourer.
Build Trailblazer for the sustain, of which Braum has none, which is ok, because he is tanky enough to withstand most of what the jungle has to offer. The Frozen Mallet is for two reasons: 1- To catch enemies who are trying to escape from being killed, or 2- For enemies who are attempting to kite you like Kalista so you can land Winter's Bite more easily. Ravenous Hydra can help to clear jungle camps and can help to clear waves in the late game, which severely helps Braum's biggest flaw, which is waveclear.
From there, you can build many different ways depending on how the game is going, my suggestion is to go into Captain Mobility Boots, Dead Man's Plate, and a Spirit Visage, which serve the purpose of sideswiping very hard into the back line of teams, engaging very hard into the front line of teams, adding to your initial attacks, and adding to the lifesteal from your Ravenous Hydra, while also adding health, which will increase the ratio of the power of your Q.
I will also attempt to lay out how to duel, 2v2, and teamfight with Jungle Braum
Dueling- Jungle Braum is a great duelist due to his stun. Most people will not be able to come even close to him because if they do, he can stun them, if they try to run, he stuns them, and if they try to kite him, he kites them. His E gives him a free move over most ranged champions and an advantage over his fellow melee champions. His ult can be used to close the kill, to keep them from fleeing, to slow them down, or in a worst case scenario, to escape from a duel gone wrong.
2v2- During a gank, you want your team to be engaged upon by theirs, this draws you in on a counter-engage. Why would this be an advantage? You see, with Braum's W and E, he is the Master of the counter gank/ counter engage, so when you hop in, and you and your laner receive your AMR and MR Buff and your shield blocks their damage, and you ult them into the air, they realize that they have made a mistake underestimating the grand ability of the jungle Braum.*
Teamfights- In teamfights, Braum can make the decision, go on the offensive, W or Flash to an ult or a stun, or you can counter engage with the same thing. Both have their merits, but you are also tanky enough to run through frontline damage, your number one priority should definitely be to go for their backline targets, and if things go south, throw up your E, and W or Flash out.*
* Important Point- Try to auto attack as many targets as possible, but pick only one to focus on yourself. While your Stun is proc'd by teammates, only you receive bonus magic damage on a target from your passive, so if you are able to stun a carry, attack them for the bonus damage.
This has been my first guide, and while not as flashy as some others, I hope it serves someone somewhere a purpose. Thank your for your time, and always remember...
The Heart is the Strongest Muscle