There's lots of references in Deadpool to the character's desecration in Origins: Wolverine (particularly in reference to sewing up his mouth), but this one's a bit more subtle. When Wade says to Weasel he became a mercenary to "travel to far-off, exotic places, meet new and exciting people, and kill them", he's actually quoting the Origins version of the character verbatim, a more subtle way of getting across that that atrocity is in the past.
In the comics, Deadpool got his name from a betting pool various scientists working on Weapon X had about which subjects would survive. Dead Pool, geddit? The film both honours and twists that - there is still a "Dead Pool", but it's in Weasel's bar, not the Weapon X base. Still, better than how they justified it in Wolverine.
We're introduced to the Dead Pool when Wade tries to get a character known only as "Buck" killed. While in the movie the only other name he goes by is "Fat Gandalf", this character appears to be a movie version of Bullet, a covert government agent and once Deadpool villain who later fought the X-Force (hence how he can appear here). As well as having the same profession, both have a similar build.