Pros:
Tanky
Good CC
Can counter jungle due to good cc, but usually requires adc and support to help.
Can adjust to being counter jungled
Good Positioning
Mustache and Poros
Movement speed with e
Gives Mr/Armor to an ally
Okay sustain
Can give red buff
Great at stealing dragon (Dragon slayer Braum for a reason) he got the massive CC to stop the enemy jungler and can flash in, steal drag, then jump to an ally.
Cons:
Weak Jungle Early Game
Relies on ally's to do damage
Takes skill to play
Needs ally's to help him position himself

For Jungling the best route is:
Gromp - Smite
Blue Buff
Wolves - Smite
Raptors/chickens
Go back to base and upgrade jungling item to chilling smite one
Krugs - Smite
Red buff
Raptors/chickens - smite
GANK!!!!
If you are low at any time during jungling I suggest getting river crab because it gives vision and hp. Additionally it feels as if not enough junglers are getting river crab, so this is an advantage from Braum to get it when possible.
If you are countered jungle/die due to being counter jungle:
Focus on the small buffs and ward behind the walls, usually the enemy jungler will wait for you at the big buffs like blue and red, so this way the enemy jungler is either wasting time waiting for you or they will return to their own jungle. I recommend getting scuttle crab if you are counter jungle to help prevent future couner jungles in the game. It is also important to ping ally's to come if you see an enemy in your jungle and always smite wolves because it is free vision in your own jungle.
Tips:
Always try to recall with enough money to buy your item and some extra pots so it will help you sustain in the jungle.
Stand near a wall where you can flash over so you can escape in case an enemy shows up by flashing over if you are low on health.
Red buff gives health back to the user, use this to your advantage when fighting an enemy for red buff as this will help you sustain longer then the enemy.

Braum requires communication with his team so he can jump to them. This is very important because it can save their/Braum's life because Braum needs the ally to be in his range of jumping. His w can help him steal drag and escape the pit, it can be used to position himself in a team light or go defend the adc, but it relies on his ally's staying near him.
Braum by himself does a fair amount of damage with his q, if he gets health tanky items such as warmogs, otherwise he relies on the team to do the damage. Braum's passive also involves his team mates helping to activate it. Even though it is simple to help Braum activate it, it is essential during team fights because stunning the main source of damage (like the adc) reduces most of the damage your team and it helps open an opportunity to burst that adc down.
Braum's shield is known for saving lives, so use this to your advantage during a team fight, as it can block many projectiles (such as ez ult). I highly suggest using it in the team fight then before it starts (unless it is to prevent a massive amount of damage or cc such as name ult). It is best during the team fights because most people tend to burst all their abilities/projectiles during the battle and not focus on Braum's shield being up.