He also sought to minimize future attention on the case.
"My comments are certainly not in proportion to the unfortunate and sad situation that we have here, but I've been advised to address the subject once, and it's time for the New England Patriots to move forward," Belichick said. "Moving forward consists of what it's always been here to build a winning football team, to be a strong pillar in the community and be a team that our fans can be proud of.
"I'm not trying to make this story disappear, but I respect the judicial process and have been advised not to comment on ongoing legal proceedings. I'm advising our players to do the same things."
The Patriots cut Hernandez after he was arrested the morning of June 26 and before he was charged that afternoon. Hernandez has pleaded not guilty.
Players are scheduled to report to training camp on Thursday when quarterback Tom Brady and the five other captains from last season will be available to reporters. The first practice is scheduled for Friday.
Belichick opened his 22-minute news conference by speaking from prepared notes.
He expressed sympathy for the family of shooting victim Odin Lloyd, said the team's in-depth process of studying a player's background is "far from perfect" but wouldn't be overhauled, and took responsibility for bringing people to the team.
"The hundreds of players we've had through this program in the last 14 years, there's been a lot of good ones, a lot of real good ones," said Belichick, who became coach in 2000. "We'll try to do a good job in bringing people into this organization in the future and try to learn from the mistakes that we've made along the way, of which there have been plenty."
At about the same time as Belichick was speaking, Hernandez was in court for a probable cause hearing in which prosecutors asked for more time to present evidence to a grand jury. The hearing was rescheduled for Aug. 22.
"Our thoughts and prayers are with the family of the victim and I extend my sympathy really to everyone who has been impacted," Belichick said. "A young man lost his life. His family has suffered a tragic loss and there's no way to understate that."
Belichick was unusually expansive in his answers. He declined to answer some questions about Hernandez, saying he had been advised not to answer those about people involved in the legal case.
"This case involves an individual who happened to be a New England Patriot, and we certainly do not condone unacceptable behavior and this does not in any way represent the way that the New England Patriots want to do things," he said. "As the coach of the team, I'm primarily responsible for the people that we bring into the football operation."
He didn't say, in response to a question, if he had talked with Hernandez since the player's name was linked to it.
Belichick said "the fundamentals" of the Patriots' player evaluation process will stay the same as they've been since he became coach in 2000, but the team will work hard to do it better.
Hernandez dropped to the fourth round in the 2010 NFL draft because of character issues. Several teams took him off their draft board.
"We look at every player's history from the moment we start discussing it," Belichick said, "going back to his family, where he grew up, what his lifestyle was like, high school, college experiences. We evaluate his performance, his intelligence, his work ethic, his motivation, his maturity, his improvement and we try to project that into our organization on a going-forward basis."
Belichick wouldn't answer a question about starting cornerback Alfonzo Dennard, who was charged with first-offense drunken driving after being pulled over in his car on July 11 in Lincoln, Neb., while on probation. He remains on the team.
Players are evaluated on "a case-by-case basis," Belichick said. "Whatever the circumstances are on any one individual, you'll have to make the decision based on an individual basis."
With Hernandez, he said, the team "acted swiftly and decisively" to cut him. "Having someone in your organization that's involved in a murder investigation is a terrible thing."
Patriots owner Robert Kraft has said he was "duped" by Hernandez. When Belichick was asked if he also had been "duped," he said he couldn't comment.
"We stress high character and we stress making good decisions," Belichick said. "We'll learn from this terrible experience that we've had.
"We'll become a better team from the lessons that we've learned."