DC Comics' Justice League has typically featured the publisher's biggest superheroes, making them one of the best, strongest, and most popular teams in the DC Universe at large. However, that's not to say that they haven't had their changes in leadership, membership, and team dynamics over the years. Fluctuations to the roster have happened, though not always for the best. A great example would be the Justice League of The New 52. While they initially had a core team of seven featuring the DC Trinity, Flash, Green Lantern, Aquaman, and newcomer Cyborg, the roster did shift after DC's Forever Evil, as the big hero of that event was surprisingly none other than Lex Luthor, who leveraged his fame and popularity to weasel his way onto the team, even attempting to take control as well. However, he may have been one of their worst members and leaders ever.
After Forever Evil, public opinion was massively in Luthor's favor, with the world generally seeing him as a hero. Lex saw himself in this new light as well. He also received a large shift in his perspective, realizing that there were much bigger threats to the Earth than Superman. The Crime Syndicate of Earth-3 traveled to Earth-1 after their own Earth had been destroyed by a malicious cosmic entity (later revealed to be the Anti-Monitor) which greatly concerned Luthor. As a result, Lex decided to build a new Watchtower in orbit above the Earth as a gift, requesting to join the League in order to make them stronger so they might be better prepared for what was no doubt coming for them in the near future.
After some leveraging and manipulation by Lex, the heroes agree to induct Luthor into the League in Justice League #33 from writer Geoff Johns with art by Doug Mahnke (if only to better keep an eye on him and his ego). However, this would prove to cause a plethora of problems. Even before his official induction in his first team-up with the League, Luthor tried to take matters into his own hands when the League was trying to contain Jessica Cruz and her ring from Earth-3 that had just chosen her. Luthor tried to assume control and lead, ordering the League to handle the attacking Doom Patrol while he would supposedly save the day and neutralize Jessica. However, it's quickly revealed that he would've only made the situation worse. Thankfully, Batman was on the scene to empathize with Cruz and calm her down, thus mitigating her power output (Lex was going to try shocking her brain, potentially frying it). In the aftermath, The Dark Knight is quick to reframe Lex's perspective, completely shutting down his aspirations for leadership with the League.
Even after Luthor was an official member, the problems didn't cease. For example, Lex still had armories and labs full to the brim with deadly weapons of all sorts, including the deadly Amazo virus which infected nearly the entire east coast within a matter of days. Not only that, but Luthor's ego remained a constant issue that put him at odds with the League and their efforts, specifically with Batman and Superman.
While DC's Rebirth event did prematurely end (and slightly retcon) the back part of Lex's tenure as a hero among the League, Luthor is actually a part of Dark Nights: Death Metal's current Justice League featuring Detective Chimp, Hawkgirl, Cyborg, Starfire, with Nightwing as their leader. However, he's joining them due to his desire to redeem himself, as Luthor is largely responsible for the darkness that has taken over the Multiverse. This is why Lex Luthor was one of the worst members of the Justice League, let alone leader: he's often the cause of the problems the League ends up having to stop.