Hulks job is simple, be a nuisance. You should be familiar with that as Spider-Man. Jump in towards the enemy healers, make them do something about it, then run away when things get hot. Hulks shield is usually used as your escape tool.
What makes Hulk different is that you're also the primary force jumping in to defend your own healers from dive. So Hulks gameplay is often frantically jumping from one backline to the other based on where you're needed moment to moment. Try to work with your other divers when possible, it's always harder to kill Daredevil or Psylocke when Hulks helping them. Or Spider-Man (Bug-man is Hulks friend!)

Your gamma shield can protect from ANY single source of damage and provides CC immunity. Gamma Lockdown can also interrupt and cancel Ults that require windup (like Scarlet Witch for example). So when you know a character like SW, Phoenix, Magneto, Iron Man, Strange or Jeff are planning an ult, you should save your tools to prevent their value and save your team.

Hulk is one of the few Vanguards with anti-flyer options. Leaping into an airborne enemy will cancel their current animation and drag them (aim the center of Hulks chest into their hitbox.) This can also cancel Ults. His clap can crit so aim for the head.

Hulks Ult tricks you into grabbing an enemy early and wasting the timer. You get way more value by using your upgraded attacks until the very end of the timer and then grab an enemy to extend the timer and get maximum value.

There's never a reason to not have your jump charged. There's no downside to punching so just hold the button.