Regarding strategy for the OP, I have some opinions on PvP setup. Again. opinions. Going down the helix:
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Persistent Projectiles. Lets you crank out one additional homing rocket, or set up a better angle for a curved shot, both of which usually land you more homing chase kills.
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Wind Chill, to discourage melee rush, slow targets in close combat, and slow major minions to control space.
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Use Ready Rockets if you aren’t using reload speed gear, and Speedy Delivery if you are. The main reason to take Ready Rockets, in my opinion, is that sometimes Speedy Delivery can make homing rockets move too fast, and they’ll hit objects before the homing effect arms.
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Rapid Reload. You don’t really need Rapid Reload with the cancel, but Murder of Rockets locks you into a predictable three-rocket firing animation. You want to be able to quickly fire Hawkeye with a regular rocket chasing it for burst damage kills.
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Player preference. Both Glide-iator and Tailwind help you stay unpredictable in the air, but most of the good Benedicts I’ve played only go airborne when they need the high angle shot, because it can leave you exposed to projectiles. Divebomb can be useful for both chase and escape, but poor aim can easily get one killed. I like Glide-iator, to open up additional trick jumps, and to get as high as possible over the heads of melee.
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Heavy Bombardment, for Hawkeye bursts.
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I get more mileage out of Danger Zone, aiming at the ground near enemies for guaranteed DPS. If you’re confident in your aim, Surgical Strike can lead to more kills, but can hurt minion management. Multiloader trades off a lot of your fire rate consistency for a potential burst, and is hard to recommend.
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First Class. Helps make Liftoff escapes from melee more successful.
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Party Starter, for Hawkeye bursts.
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Phoenix Protocol, when you get it. Use it like OMs napalm, force enemies to retreat into the AoE, or just slam it into a team position to open combat. Until the unlock, Rockets Launchin’ Rockets isn’t terrible, but you can’t fire the smaller rockets right after Boomsday activation, and you need some distance from your target to get all the small rockets to fire. I typically fire Boomsday in the opposite direction of my target, into an open area, then do a midair 180 into the final approach.
Regarding general combat tips, I like to adapt strategies from TF2 Soldier play. Treat Liftoff like a rocket jump, get in close, fire at feet or heads. When you drop an enemy’s shields and poke their health a little, chase Hawkeye with a regular rocket at their feet, point towards the sky, and start arcing the shots down onto heads for potential crits.
Benedict clears minions fast, even faster with Danger Zone, and it’s easy for him to shift focus. It’s typically worth burning Hawkeye or Boomsday to quickly wipe a minion wave.
Boomsday is good for minion clear and softening targets, but chasing weakened opponents can often be a waste of time, especially if they find a healer. Beware of enemy Pendles–a good snek will be waiting for your Boomsday activation for a free kill. If you know the ambush is coming, you can 180 the rocket back at your feet with Phoenix Protocol, and usually out-DPS him.
For Gear, a bit of Reload speed to set up smooth reload cancelling is desirable. UPR Nimble Turtlegard Armor (DR, Reload) or Nimble Breach Kit (Shield Pen, Reload) combined with the AGM Loader (Lore Legendary) is common, plus a 0-cost shard extractor to get things up fast. If you don’t have the Lore item yet, or if you’re playing in a tournament and legendaries are banned, Rifleman’s Hailfire Mags (Double Reload) is a solid substitute. Regarding other legendaries, I’ve seen Alamo-7, Bunker Buster, and Vyn’s Quiver used a lot. Combined with DR gear, the health regen of Vyn’s can make Benedict pretty tanky.
Some character-specific stuff: When you get around level 8 or so, a good Hawkeye burst does enough damage to put a Galilea player at risk of shield break. At level 10, you can even try breaking a Boldur shield, but there’s still a ton of HP to go through after that.
Enemy paper tiger Battleborn are your friends–Orendi, El Dragón, and Rath especially. Don’t be afraid to engage them close at the start of the round, especially once you’ve hit level 2 and gotten the slow on Liftoff. Phoebe also has a hard time with Benedict, and it’s important to keep her under pressure and under-leveled, even if she just keeps retreating into Phasegate.