On Monday, Edwards was booked by police for driving 38kph over the speed limit and refusing to undergo a second test after alcohol was detected on his breath.
The club suspended the 24-year-old until round five, including the pre-season competition, and has also fined him $5000.
Brayshaw, currently in Perth, is yet to speak with Edwards but plans to sit down with him early next week to discuss the gravity of the indiscretion.
"Aaron's just got to understand that that sort of behaviour's not going to be tolerated by this footy club," he said.
"It's not acceptable to be 38km over the limit. It's not acceptable to refuse a blood test. It's been made very clear to him.
"He's got to take what's been handed to him very seriously and understand that his career's on the line here. The penny's got to drop with him."
Edwards has played 26 games for the Roos in two years and would have featured in more if not for a broken leg that sidelined him for most of 2008.
His aerial strength and attacking presence shone in North's run to the 2007 preliminary finals and it looked set to continue last year.
Edwards has also settled in well with his teammates at North after starting his career with West Coast and spending 12 months in the VFL.
"He's a terrific bloke, I really like him," Brayshaw said. "He's a very good footballer who we want playing for our footy club, but there are certain behavioural standards attached to playing for North Melbourne.
"His behaviour recently fell well short of that."
Brayshaw was pleased with the reaction of the club's leadership group, as new captain Brent Harvey dealt with the public punishment of a teammate for the first time.
He also supported the penalty that Edwards was handed.
"He (Harvey) stood up and handled it really, really well with Drew (Petrie) and Daniel (Pratt) and Brady (Rawlings) behind him," Brayshaw said.
"I thought it was decisive ... they handled as I understand the discussions with Aaron really well and that's why we've got these people in place.
"They're there to do those sorts of jobs and they don't enjoy it. Disciplining your own teammate is not something people would like to do every day. When [they had] to do it, those blokes stood up.
"What Aaron's got to understand now is that he's got his career in his own hands. He's got to make sure that we don't see any of this behaviour in the future."
Edwards was also reprimanded by the club for drunkenness at a Lionel Richie concert in 2007.
