No, they are not all considered part of the same canon anymore.
According to Lauren Schuler Donner, one of the producers of all of those films, you can now safely consider X-Men 3: The Last Stand and X-Men Origins: Wolverine as non-canon. She explains this at the premier to X-Men: Days of Future Past:
“Just forget about ‘X3’ and the first ‘Wolverine’ - forget about that, too!“
That means things like, e.g. Professor X's death, Logan and Victor Creed being brothers, and the original Deadpool are no longer canon.
As far as the continuity between the remaining films, they all appear to fit into a single coherent continuity, but it's a bit complicated, partly because there is time travel involved.
X-Men and X-Men 2 follow neatly one after the other. Things that were set up in the first film (Senator Kelly's character, Wolverine's past) are paid off in the second.
First Class is actually a prequel, occurring at a much earlier point in time. However, it does make reference to plot elements from the first two movies, particularly the fact that Charles and Erick were initially friends, but we also see a young Stryker make an appearance and become fascinated with mutants. (There are issues in First Class with characters that based on the comics don't make sense; I think Havoc is way too old to be Cyclops' brother, so he just isn't, things like that. I don't think there's any real continuity problems just within the movies.)
The Wolverine is almost entirely independent of the previous movies, though it clearly occurs after X-Men 2. The final scene in that movie calls back to X-Men 3, with Wolverine being surprised that Xavier is alive; but they never explain how that happened, and now that X-Men 3 is non-canon they don't have to.
Days of Future Past acts as a bridge between the two different movie eras. We see the both the original X-Men and the First Class X-Men co-existing in the same movie, with a time-travelling Wolverine acting as the connecting point.
Days of Future Past also effectively rebooted the series, with Wolverine changing the past and altering the status quo of the "present time" into something a bit more upbeat that it previously had been. This was intentional on Fox's part; as Donner points out, Days of Future Past was partly trying to "fix" all the dumb things that happened in X-Men 3.
Future movies, such as the upcoming X-Men: Age of Apocalypse, seem to be set in this new post-Days of Future Past timeline, though they don't necessarily occur chronologically. Age of Apocalypse, for example, will be set in the 1980s, about 10 years after the end of Days of Future Past and 20 years before X-Men and X-Men 2, and show the evolution of many of the X-Men characters into their modern X-Men versions.