Marvel's Avengers and DC's Justice League aren't easy groups to join if you don't have superpowers. While they've let in some odd members from time-to-time, usually, to get into either super organization, a hero needs to make a compelling case to back up their abilities. One Marvel Comics hero actually has actually been a member of both the Avengers and the Justice League: Hawkeye.
Clint Barton aka Hawkeye first appeared in Tales of Suspense #57. The character's abilities (and lack of powers) have stayed pretty consistent over time, as Hawkeye is a talented archer who doesn't miss many shots. His lack of powers has made him the butt of many jokes over the years, but the fact he's been a core member of the Avengers since the mid-'60s is impressive. What might be more impressive is that Hawkeye was technically a member of the Justice League, too.
JLA/Avengers by Kurt Busiek, George Perez, Tom Smith, and Comicraft, is one of the more ambitious crossovers that Marvel and DC have ever done. In the book, a villain name Krona is tricked by the Grandmaster to get the Avengers and Justice League to assemble the most powerful weapons in each universe. Ultimately, reality experiences a number of changes. Issue three opens with the Justice League and Hawkeye at the Source Wall, reveling in the fact they defeated Doctor Doom for good. Black Canary tells Hawkeye she's glad he left his Earth and joined the Justice League as the two share a kiss. However, a cosmic glitch soon makes the archer disappear.
The rest of the Justice League are completely confused when Green Arrow expresses his own confusion about how none of the team remember Hawkeye was there before the glitch. Even Black Canary immediately comforts Green Arrow. While we don't get to see Hawkeye on a mission as part of the Justice League, in this story, he did leave the Avengers to join his DC counterparts and assisted them in taking down Doctor Doom (the universe was a mess thanks to Krona). He was a member of the Justice League - even if it was briefly.
It's too bad the universe (both Marvel's and DC's) was playing tricks on Hawkeye, because the next time a member of the Avengers tried to make fun of him for not having powers, he could have easily responded with "yeah, well have you been a member of both the Avengers and the Justice League?" Sadly, per usual, Hawkeye can't catch a break. It would have been cool to see another archer join the Justice League - although, Green Arrow was understandably jealous. Still, it's neat to know that at least one hero has been a part of both the Avengers and Justice League.