In the movie when Ajax was explaining to him about how they were going to torture him next, he mentions that Wade was dishonorably discharged and there does not seem to be much information on what the actual reason for him being so.
Ajax mentions this around 0:42
Although he seemed to have been an assassin, as stated by Marvel Universe Wiki
Following a brief stint of military service, Wade began his mercenary career while still in his late teens. Accepting assassination jobs only against those whom he felt deserved death, he made a habit of using plastic surgery and technology to take on a new identity whenever he failed an assignment.
Why was Wade dishonorably discharged from the Army?