The former Australia international, who now has British citizenship, lead Queensland to five Sheffield Shields during his time in charge of the state.
"I've been around Lancashire nearly six years now and it's a great honour just to be asked," he said.
"I was offered the job before but it maybe wasn't the right time.
"I'd just come back from nearly 12 years with Queensland and I felt I just needed a break.
"Now I'm ready to grab it with both hands and hopefully win something. It feels like we've been on the verge of something special for the last two or three years."
He added: "The captain is the one who makes the decisions but there should be 11 captains out there making choices and trying to get the game done and dusted."
Law had considered retirement at the end of last season when Lancashire narrowly failed to end a 73-year wait to be crowned outright county champions on the final day of the campaign.
But after signing a one-year extension to his deal and accepting the chance to succeed Chilton, he insists he is as hungry as ever.
"I've taken this on after discussion with Jack (Simmons, chairman) and Mike (Watkinson, cricket manager) about my playing future.
"We talked about blocking out a series - maybe the Twenty20 or the Pro40 - but I went away and I thought 'I can't play cricket like this, not giving 100 per cent in every game' so I came back and said 'I want to play in every game possible'.
"I'm putting my name on the team-sheet for every game I can. That's my priority. For me, not getting the most out of myself would be letting the team down."