Here’s the build I use for PVP. For PVE, I swap out Bullet Buff for Icy Resolve to get the extra DR.
Gear wise go with attack speed, max health, health regen, and damage reduction. I would highly recommend his legendary because it has AS, max health, and helps with keeping his gun cool (you can’t fire forever, but you can fire longer). Remember that AS gear will increase your rate of heat accumulation, but The Cooler and Barrel Cooling will help offset that. Still, you should fire in bursts as much as possible unless you’re helping a teammate or fleeing from an enemy BB.
Just remember he’s a Defender. Your job is not to pursue enemy BB, it’s to drive them away, help your teammates kill those who are vulnerable, knock enemy BB off of your teammates, kill minions, and build stuff. Don’t expect to have a high kill count, but you should have a respectable assist count. At the start of the match, pick a spot behind the front line with convenient cover you can dash behind when things get too hot. When your team is ready to push, stay in the middle of them and protect your support and snipers. Monty is not a pusher and should never lead the charge unless you have a pocket healer like Miko or a good support like Alani and/or Reyna. If you have a melee or or an Oscar Mike, you should be right behind them shooting over their heads and keeping the enemy from focusing on them. Always be aware of your surroundings so you can dash to safety if need-be and stay away from both Shayne and Aurox and Ghalt because both of them can ruin your day with their fetches. If you’re attacked by a melee character, try to get their backs to a wall and dash them into it and then either finish them off or run depending on the situation (1v1 with most of your health, kill 'em. 2v1 or low health, run unless you have some backup).
Monty is a lot of fun to play in PVP, but you have to be careful because he’s such a big target and can be taken down quickly despite his high health pool if he gets focused.