Who are the members of Suicide Squad? Is Joker a member of Suicide Squad?
This page isn't of much help.
No, he is not.
The Suicide Squad is a group of "minor" villains from the various DC comic lines (many of them were Batman, Flash, or Green Arrow villains). There have been several incarnations of them, but the best-known is the most recent one run by Amanda Waller. She takes villains that are currently incarcerated and sends them out on dangerous missions, considering them "expendable".
The entire list of members is extremely long, but none of the "A-list" villains were ever members. That means no Joker, no Lex Luthor, etc.
As far as the upcoming movie, the trailer has a number of scenes showing the entire squad together, and the Joker is not in them. There are also a number promotional videos, like this one, showcasing the Squad members. The known members of this incarnation of the Squad are:
In addition, Amanda Waller, her "team captain" Rick Flag, and Flag's bodyguard Katana are also part of the team. Note, though, that none of these three are criminals, so they don't have the same "suicide switch" that the rest do to keep them in line.
In the movie (and in their comic book series), the Squad are the protagonists, and are generally portrayed as anti-heroes: they act like villains, but they are doing good work and helping people in the end. The Joker, on the other hand, appears to be acting as straight-up villain. He may be the main antagonist of the movie (no other villain is shown in the trailers), or he may just be keeping tabs on Harley, which would fit with his comics persona, it's not clear yet.