Incoming big post of recommendations! Cause ya know I can't just list numbers without talking about the stories themselves, haha. Better warm up the notepads now, this is gonna be pretty long...
Black Widow: Homecoming by Richard K Morgan. This is my favorite Widow series/story and it delves her past and the Red Room a lot. The things that were done to her are pretty messed up, but fascinating and Natasha never falls into either self-pitying or self-loathing. The art is also gorgeous in this story, haha. The same creative also did a sequel called The Things They Say About Her, but I don't find it as good. Still, entertaining though and it has Daredevil show up, if you're a DD fan.
Black Widow: The Name of the Rose by Marjorie Liu. I haven't actually finished reading this yet, haha, but its really good from what I've read so far! Unlike Homecoming, which only had one cameo from Fury, this has quite a few Avengers popping up as Natasha has been accused of spying on them and selling their secrets. It's still largely about Natasha going solo though and she gets to fight her way through some tricky situations. But, I would highly suggest you read Homecoming first since it covers her history and something she learns in Homecoming explains something else in Rose.
Hawkeye by Mark Gruenwald. One of my stories/Hawkeye/Mockingbird stories. With Gyrich running the Avengers there wasn't a spot on the roster for Clint and he's forced to take a job as a security chief to the pay the bills since he can't live for free for the Avengers Mansion now. The interplay between Clint and Bobbi is fun, even though Clint spends a lot of the story depressed because a betrayal early on. It digs into Clint's insecurities quite a bit with Crossfire deliberating targeting him because he perceives him to be the weakest of the superhero community and a brief cameo from Cap has Clint feeling jealous despite Bobbi shameless hitting on him from the second they meet. But it doesn't wallow in the insecurities too much and there's some fun fight scenes throughout.
New Avengers: Reunion, Hawkeye & Mockingbird, H&M/Black Widow: Widowmaker, Blindspot all by Jim McCann. McCann isn't without his flaws, but his stories show a pretty complex Clint. Most of the time you'll only get to see one or two sides of Clint just cause it's hard to fit in all these sides to him within a story. But because McCann's stories are all pretty personal and emotional as well as fun action adventures there's plenty of chances to show different sides to him. At one point or another you'll see Clint as the nice guy, the hot head, clever, stubborn, funny, cocky, insecure. So, as a Hawkeye fan I can really appreciate that. Also don't hurt he writes a pretty kick ass Mockingbird, haha. Like Clint, McCann also gives her some very personal stories and Bobbi really gets put through some tough stuff, which doesn't leave her in the best place mentally. It's interesting to see her try to deal with her time held active with the Skrulls and how her detached personality can really make things hard on her. Anyway, read em, haha.
Avengers 137 - 149. This is several stories which can be found in the Coming of the Beast and Serpent Crown hardcovers, but I'm just gonna lump em together since it's just one straight run of issues. From issues 137 - 141 you get a pretty cool story about Hank and Jan returning to the Avengers, but Jan gets hurt and Hank's powers growth powers aren't stable which leads to him comatose though still growing. Vision has to use his phase powers to go inside growing body, which is pretty cool stuff and a neat twist on the usual shrinking down to go inside someone stories. It's also the introduction of Beast to the Avengers and sets up the Cowboy Clint time travel story. The Cowboy Clint story is pretty short and goes from 142-143, but it's so much fun with Clint getting to live out his dream of being a cowboy in the old west. It still cracks me up he runs around shirtless instead of buying normal clothes cause people are weirded out by the superhero gear. 145 -146 is another short story about an assassination attempt on Cap where he's left clinging to life and you get to see a little bit of how much he means to the Avengers as the various members deal with the possibility he might die. It also has a surprisingly dark ending for the bad guys. The rest of the issues have the Avengers fight an evil version of what is super obvious the Justice League, haha. I found this to be a really endearing stretch of issues though, like how happy Jan is to see Clint and the other Avengers again, the little scenes between Hank and Jan both laid up in the hospital together, characters trying to support each other when dealing with Cap's grave injuries.
Avengers: Korvac Saga -- When people talk about best Avengers stories most of the time it comes down to Kree-Skrull War and Korvac Sage. So, yeah, read Korvac Sage, it's pretty good, haha. You should start with Thor Annual 6, since it introduces the Guardians of the Galaxy ( a very different team than the movie line up) and then move on to Avengers 168, skip 169 which is a filler issue, and read from 170 - 177. I particularly like 174 which is a pretty awesome issue with Clint taking on the Collector single handedly after the other Avengers have been captured. There's also another nice Clint moment in an earlier issue when he tries to cheer up Wanda after Vision and Pietro go missing. But asides from my bias, it's a great and has a lot of Avengers showing up like Jan, Hank, early Ms Marvel, Hercules, Wonder Man.
Avengers: Kree-Skrull War -- The other best Avenger story, haha. This stretches from 89 - 97 but I would throw in 99 - 100 too since it follows up on what happens to Clint at the end of the story. I actually haven't read this yet, because I wanna get the Masterworks hardcover which is the only collection that has the Clint follow up stuff, but that goes for like 45 bucks at the cheapest. I'll get it one of these days!
Avengers: Under Seige - This takes place from 270 - 271, 273 - 277. This THE Masters of Evil story and inspired the EMH episode of the same name. The comic version is a lot more brutal though, with Jarvis and Black Knight suffering bad injuries. Zemo also has a great villain moment where he finds Cap's only pictures of, I think, his parents and just rips em up right in front of him.
Avengers Annual 16/West Coast Avengers Annual 2 - This is a pretty awesome story and well known as the time Hawkeye conned Death (technically the Grandmaster with Death's powers). This has a lot of cool fights with the two Avenger teams facing off, a great fight with Hawkeye vs She Hulk, Hank vs Jan, Bobbi vs Cap. Bad guys get involved too, of course, and only Cap and Hawkeye are left with Clint saving the day cause he's a cheater, haha. Even if you aren't into heroes fighting you should check it out!
Solo Avengers 1 -5. Although Clint's pre and post Swordsman past has come up recently in McCann and Fraction's stories, these were the first issues to get into that part of Clint's life and the present day story with Trickshot makes it well worth reading. It's about a bounty being put on Clint's capture which gets him in all sorts of trouble, fighting off bad guys before he eventually gets knocked out and wakes up on this foreign island with no weapons, no clothes, traps all around, and Trickshot challenging him to beat him. And then ya get a great twist at the end about why Trickshot would go after Clint now, after all these years.
Marvel team up 95 - A pre- Avengers Mockingbird teams up with Spider-man to take down corruption within SHIELD. What more do I gotta say? A fun little story to check out when you aint in the mood for a long epic.
Original Thunderbolts run 1 - 75, plus Thunderbolts Annual 2000 - I haven't read all of these cause Marvel doesn't wanna put out any trades, but what I've read I really liked. Busiek and Nicieza are both great writers and it's got Hawkeye so you know I'm on board, haha. Clint joins in issue 21, if you're curious, and 22 has a great story where he takes on Hercules in defense of the T-Bolts cause Herc is mad at Atlas for injuring him when Atlas used to be bad. The annual takes place before #38 and it's about Clint going to Mephisto's realm because he believes her soul is trapped there. For context on how he finds that out ya need to read Avengers 10 -11 where dead Avengers are revived as zombies to fight the living and as she is being put to peace Bobbi says to tell Clint something, but gets caught off before she can finish.. So when Clint hears of it, he rushes to Mephisto's realm, intending to face Mephisto's mind games by himself in a bid to save Bobbi, and is surprised when the rest of the T-Bolts come after him to help. Then Thunderbolts #38 follows up on the story a little bit, with Clint briefly wondering if he's tainting Bobbi's memory by dating Moonstone. Thunderbolts 50 has a pretty interesting turning point in the series that I'll talk more about below.
Thunderbolts: Life Sentences - This is a really, really good issue. Like, I would say read it before anything else just so ya don't gotta wait, haha. But in T-Bolts #50 Clint agrees to go to prison and in return the Thunderbolts are pardoned for their crimes, but they have to disband. So, this one shot or annual or whatever ya wanna call it is about Clint trying to deal with being in prison surrounded by people that literally try to murder him and Songbird, Moonstone, and Mach IV dealing with trying to be normal people. Clint quickly agreed to go to prison, thinking he'll figure it out like always, but is hit hard by the reality of the terrible situation. I don't remember Songbird and Moonstone's story quite as clearly, but they're really good too. But it ends with Clint agreeing to help a couple other prisoners break out and #51 onward has an ongoing story for a while about Clint being on the run with the prisoners and Songbird being tasked by SHIELD to track him. There's another plot going on too with the other T-Bolts members but I cant't remember what it is right now, haha.
Avengers/Thunderbolts by Kurt Busiek and Fabian Nicieza - We talked about this one before, Rogue, so I won't get too into describing it. It takes place after T-Bolts #75 though, which is when Clint leaves the book and is back on the Avengers. And yeah, it has that fun conflict of of the rest of the Avengers believing the Thunderbolts have fallen back into their villain habits and Clint wanting to back up his old family, but also feeling an allegiance to the T-Bolts after mentoring them for so long.
Kang Dynasty, Ultron Unlimited, Avengers Forever. The best of Busiek's Avenger run, though you can also just read all of it and that's good too, haha. Ultron Unlimited is 19 -22 and is pretty awesome Ultron that birthed the popular "Ultron, we would have words with thee" phrase that you hear Thor repeated in EMH. Kang Dynasty is 45 - 56 is an epic story about Kang succeeding in defeating the Avengers and taking over the world and the Avengers having to claw back to defeat him. There's a special "nuff said" issue as part of a company wide thing where no words would be spoken in any of the comics and Busiek handles it beautiful. The issue is where Kang wins and the lack of words adds a lot of weight when Kang specifically makes Wasp signs the paper surrender agreement and when Thor looks upon a vaporized Washington DC and falls to knees at the sight of so much death. Avengers Forever is a 12 issue epic that plucks Avengers from different points in their history so I would suggest you probably wait a little while before checking that one out, just cause you'll enjoy it more.
Avengers: Earth's Mightest Heroes by Joe Casey. Not to be confused with Avengers: Earth's Mightiest Heroes by Chris Yost, which is based on the cartoon. There are two volumes and the first is set around the origin of the team and when Cap is first unfrozen, but instead of rehashing the origin it's about the stuff you didn't see. Like it shows Cap visiting the Vietnam memorial wall and his reaction to the type of crazy reality shows. The second volume is set around the time Vision joins the team and when Hank is having his mental break and took on the arrogant Yellowjacket personality. It retcons some things when it comes to Yellowjacket so the Avengers actually know its Hank beneath the costume and aren't totally okay with this stranger up and marrying Jan, haha. It also has some cool stuff with Clint supporting Vision, who the public is weary of since he was created by Ultron, and some fun Super Adaptoid action!
Captain America 1-7 by Mark Waid. Waaay before he got acclaim for Daredevil Mark Waid wrote Cap in early 2000 and it was really good. 4 is a great little issue where Cap is feeling put off by America's obsession over him and Clint tries his best to cheer him up and explain why he's important to the world. 5 -7 has Cap being captured and impersonated by a skrull, who abuses Cap's fame. Bit of an inspiration for the EMH story where Cap also gets impersonated by a Skrull, though the Waid story has fake-Cap more focused on the public rather than destroying the Avengers from within.
X-Men: First Class - Cause I might as well throw one non-Avenger suggestion in there, haha. This is about the actual first class, not the movie version, and it's pretty fun. The original five had great chemistry together, it was a lot of fun seeing em together again set in simpler times.
One last suggestion that i'm not sure Marvel Unlimited will have, but I gotta suggest. Mini Marvels. You'll laugh so hard at these, ya gotta check em out.