While Wade Wilson was turned into a character vaguely similar to Deadpool in X-Men Origins: Wolverine, that specific story has been effectively undone by the events of Days of Future Past.
The new Deadpool movie does, however, exist within the post-DOFP x-men-cinematic-universe. Producer Simon Kinberg recently explained this:
But I work on all of those films in one capacity or another, either as a producer on all of them and as a writer on Fantastic Four and this movie, so I'm certainly aware of all the different stories we're telling at the same time, and they all are part of a larger fabric now, and so the world of Deadpool, the world of Gambit, exists in a post-X-Men: Days of Future Past post-X-Men: Apocalypse world where all of these stories are the same as our shared history. The same way that each of us of different ages knows about Nixon and knows about Reagan and knows about 9/11, our fictitious events like the stadium dropping on the White House in 1973 is part of the world in which Gambit, Deadpool, Wolverine 3 on forward exists.
In the movie, Deadpool visits The Xavier School for Higher Learning, and interacts with Colossus and Negasonic Teenage Warhead, both members of the X-Men. Additionally, Colossus repeatedly attempts to recruit Deadpool to the X-Men team.
More importantly, Deadpool is revealed to only have more recently gained his mutant abilities as an adult, as a result of experimentation, long after the events of X-Men Origins: Wolverine.
Breaking the fourth wall repeatedly, Deadpool references the previous X-Men movies. He has an action figure of Wade Wilson from the Origins movie, and comments that it's hard to keep the timelines of the X-Men movies straight.