An important part of making a Justice League film work is creating a world in which these heroes would realistically interact. It's also important to find a way for them to realistically come together. The Avengers was able to do this by having a recruitment character. The Justice League should go a different route to differentiate themselves from the Avengers, but what should that route be?
How about stitching together side-characters from each franchise to create a cohesive, realistic universe? By connecting the side characters in a universe, the heroes are more likely to come together and interact. This method could also justify the position of each member of the team.
For example, we can combine the League of Shadows (Batman), Kryptonian technology (Superman), A.R.G.U.S. (Wonder Woman), the Red Room (Cyborg), and the Guardians of the Universe (Green Lantern) to tell a compelling Justice League story.
Let's say that after Man of Steel, the U.S. government keeps some of the technology left behind from a Kryptonian invasion (Superman). They could give it to the Red Room (Cyborg) to study and to A.R.G.U.S. (Wonder Woman) to protect. That links three characters: Superman, the hero, Wonder Woman, the secret, and Silas Stone, the creator of Cyborg. Now let's say that the terrorist organization known as the League of Shadows (Batman) infiltrates A.R.G.U.S. and the Red Room. They steal the Kryptonian tech and catch the government by surprise. To make this work, the League of Shadows would have to be incredibly upgraded to Bond villains status amd be completely organized, secretive, extensive, and powerful. Their goal would be to destroy cities that they see as corrupt and beyond saving in one fell swoop. The technology they steal could be some kind of mind control tech or space weapon or whatever you think would make the story work. Batman gets drawn in because he was a part of the League of Shadows and he knows how they operate. Superman gets pulled in by the government because it is his tech. Wonder Woman gets pulled in because A.R.G.U.S. was attacked and she's worried about protecting the Earth. Cyborg gets pulled in to save his father, who saved him, but who is now captured by the League of Shadows. The Guardians of the Universe think Earth has dealt with aliens before it has been ready to. And they are concerned by the idea of human power dynamics changing due to extraterrestrial technology, so they dispatch Hal Jordan to investigate. This brings Green Lantern into the fold.
These human bad guys are a challenge for the Justice League because of the technology they possess. While their plot is foiled, they continue to operate in secret, which Batman knows. This keeps the rag-tag team together for future missions. So the LEAGUE of Shadows leads to the formation of the Justice LEAGUE. Cyborg is able to recruit new members like Martian Manhunter, Aquaman, Shazam, Hawkman, Zatanna, Green Arrow, etc. by using an A.R.G.U.S. database that he calls "The Grid." Part of the League of Shadows branches off and becomes the Secret Society (villains for future Justice League films).
In terms of character development, Superman realizes that he can do more good with a team and that his technology could be more dangerous than he is. The latter leads him to store items in the Fortress of Solitude. Batman expands his purpose beyond Gotham to save the world by fighting the League of Shadows and the Secret Society. He comes to terms with his past in the organization. Wonder Woman makes her debut to the world and finds acceptance. She also starts to connect more with Steve Trevor. Green Lantern realizes that he hasn't been as active as he should be on Earth, which re-orients his world view. And Cyborg forms a closer bond with his father while finding a new purpose in life. A.R.G.U.S. begins to fear metahumans and secret organizations that they know nothing about. This kick starts Project Cadmus, which will drive a wedge between them and the Justice League. Lexcorp secretly finds and studies some of the stolen Kryptonian technology, which attracts Brainiac to Earth. Mid-credits, we find out that the government has been keeping Martian Manhunter frozen in ice since the 1950s. Post-credits, we find out that the government knows about Atlantis and considers it a threat.
Note: If I could think of an organic way to tie in Flash and/or Aquaman without overburdening the film, then I would. I'm not sure how to incorporate the speed force or other Flash concepts, and I'm not sure how to incorporate Atlantis without giving its own film. But I'd love to see your ideas below!
**Edit on Martian Manhunter:
There shld be this one secret agent character who's always there in the background and stuff whos maybe not vital but has an integral role,he doesnt havta be addressed or he can just be called agent X n then in the mid credits scene we see that agents badge which reads 'John jones' n his eyes turning red or sumthin,it'll fit more with the creepy secretive silent watcher MM of the new 52, n aquaman n flash need their own films