You shouldn't be putting too much stock into character issues, as far as picks for the patriots. They traded for Moss and Dillon, who both had character issues at the time and they've drafted guys like Tate, Spikes, Meriweather and even Brandon Deaderick all had some character concerns heading into the draft and the Patriots still took them. My point is is that if these guys interview well and the Patriots believe they can be productive players and help the team they will put those few character issues aside.
Moss and Dillon were interviewed prior to the Pats pulling the trigger on them.. And they had years in the league with clean noses.
Meriweather had the one on field issue. The off-field incident was him protecting a friend.
Spike had the porn tape, but that came out after he was taken and then the ADHD meds, but, again, after he was taken. If you are referring to the 2009 incident with the GT RB, Spikes got Kudos for manning up and extending the suspension from a half game to a full game so as to not be a distraction.
Tate had a positive test for POT at the combine. Beyond that, he had no issues. In fact, it was noted that he was excellent at taking coaching and was willing to put in extra time when needed.
Deaderick's is the one who had an issue, though, he was made team captain his senior year. But he showed his immaturity again with the Pats.
The Pats have shown they are willing to tolerate certain things that can be written off to immaturity if you have shown improvement since that time. But there are certain "character" issues they don't tolerate. Like an unwillingness to be coached. Or an unwillingness to watch film. Things that dramatically impact your performance on the field.
That being said, the Pats have also parted ways with problems that can become a smudge on the team. Especially since the Krafts bought the team. Willie Andrews, Christian Peter and Kenyatta Jones all come to mind.