The move to appeal has angered many who want to move on with their lives "and let Christine and Amber rest in peace", but Mr Levick said he would fight to have Lundy freed.
"I have to be an optimist. If I wasn't an optimist, I would have given up years ago," Mr Levick said.
"I'm pretty sure [the Privy Council] will quash his conviction and they will order a retrial and if they order a retrial then it's up to the Crown to decide, just like in the Bain case, whether they have a retrial or not."
Mr Levick compares the work he has done on the Lundy case over the last decade to the campaigning by former All Black and entrepreneur Joe Karam in the fight to clear David Bain's name.
Most of the public share negative views of Lundy's claims of innocence which means Mr Levick, like Mr Karam with the Bain case, finds himself on the unpopular side of the divide.
"I'm not interested in being popular," Mr Levick said. "You just look at this case and ... from the beginning to end, the evidence is codswallop.
"So, having decided that 10 years ago, having read all the evidence, I just have to do something about it. It's pretty simple."
Lundy's lawyer, David Hislop, QC, said the defence case would focus on the science used to identify DNA on Lundy's shirt as being his wife's brain tissue, which is considered the most important piece of evidence that linked Lundy to the killings.
Mr Levick accused people of criticising Lundy as being "99 per cent uninformed".
He claimed most people decided Lundy's guilt because they thought "he's fat, he put on a big act at the funeral and he used a prostitute".
"If everyone who was fat, behaved badly and used a prostitute was put in jail we wouldn't have enough jails," Mr Levick said.
He defended the drawn-out legal bid to appeal against Lundy's conviction, and said he had sympathy for the families of Christine and Amber who have to relive the trauma every time the case is back before a court.
"We're not doing anything to try and annoy them; we have no intention of upsetting them," Mr Levick said. "But on the other side of the coin ... what about the poor bastard sitting in prison?"
Christine's brother, Glenn Weggery, last week said the appeal was very tough for the family.
"It's 12 years later, we're trying to move on with our life and let Christine and Amber rest in peace and it's just not being allowed to be done," Mr Weggery said.
"It keeps getting dragged up every few months.
"He's [Lundy] got the right to appeal, fine. But it's been dragged out for so long, and it took so long for them to lodge their appeal when they've been talking about it for years.
"Did any of them ever care about Christine and Amber? Because they sure as hell don't care about their families."
In 2002 Lundy lost an appeal to the Court of Appeal and had his non-parole period increased to 20 years.
