I use him last, so these are mostly solo tips, but some quick notes for ya:
-General neutral game consists of a lot of jump back triple arrow. You can also do j.:h: then triple arrow to give your opponent another hitbox to worry about. Best way to move forward is jumping triple arrow (as close to the ground as possible) then wavedash a couple times behind the arrows. You can mix in jumping spritzers (electric shot) for a little extra chip, but it’s easier to dodge.
-Be careful of his grounded arrow strings, a lot of them put you at a disadvantage. For instance, jump back triple arrow, land, :h: shot, standing triple arrow (popular day 1 flowchart) is easy to jump over and doesn’t put you in a good spot if they block it. The only grounded arrow you should be poking with is :qcb:+:m: xx :h: (“Speed shot!”). So your general ground to air string should be jumping triple arrow, land, speed shot.
-Clint has bad hitconfirms, so unless you’re punishing a move with c.:l:, don’t be fishing for slides into launch, it’s very unsafe. Instead play the footsie game with slide xx poison tip. If they block it, it’s pretty safe, if it hits, you get about 200k added to your zoning game. Abuse s. xx poison tip as well, Yipes stated that this alone beats pretty much everything Akuma does. Standing has a crazy hitbox and is your go to, close-range anti-air.
-If you’re using him with an assist, you can get away with different grounded strings. Jump back triple arrow, land, s.:h: + call assist, up arrows of choice (depending on what the opponent is doing and how much your assist covers you).
-Probably the most important thing for solo Hawkeye is to have meter. That’s the only way you can make anchor Clint work. This allows you to kill a character with lvl three if you have to, but most importantly it gives you plenty of level ones to work with. Hawkeye’s solo zoning isn’t free of holes, so if your opponent finds their way in, let a super scattershot rip to push em back out again. But what you really need the meter for is…
-Gimlet. If you have 5 meters, you have 5 Gimlets, that’s the only way to see it. If you’re gonna play Hawkeye, you need decent reactions. If your opponent airdashes, Gimlet. If they try to zone, Gimlet. If they call an assist, Gimlet. It’s more about putting the fear into your opponent and cutting off their options, Hawkeye has good matchups against popular anchors for this reason. What’s Doom gonna do that’s not gonna get shot? Nothing.
Just some random stuff off the top of my head, hope it helps.