Educated at Foxton Beach School and Manawatu College, Packer originally played rugby union from a young age for the junior Foxton rugby club and also for a junior Marist club in Nelson.
Packer also played in the Foxton Rugby League Under 14s side at the age of nine years in a curtain raiser between the Manawatu Mustangs and Auckland Warriors reserve-grade side in 1998.
Packer then played for the Levin Lions. In 2005, Packer was named in the New Zealand under-16 side. When he was only 16 he debuted for the Central Falcons in the Bartercard Cup. He again achieved national selection in 2006 when he was named both the captain of the Junior Kiwis and as a member of the New Zealand 'A' squad. In 2007, he was named in the New Zealand under-18 side.
In 2007, Packer signed with the New Zealand Warriors but spent most of the season playing for the Auckland Lions in the NSWRL Premier League. He also played for the Central Falcons in the final season of the Bartercard Cup. At the end of 2007 he received an offer from the Sydney Roosters but turned them down, preferring to stay in New Zealand.
He made his NRL debut in Round 10 on 4 May 2008 against the Canberra Raiders. He played in four other NRL games in 2008 and also played twenty games for the Junior Warriors in the new Toyota Cup competition. At the end of the season Packer was named in the National Youth Competition's Team of the Year alongside teammate Ben Matulino. He finished his Toyota Cup career at the end of 2009 with twenty three appearances and five tries.
In 2010, Packer played for the New Zealand Māori against England.
Packer was selected to play for the Warriors in the 2011 NRL Grand final against the Manly-Warringah Sea Eagles as a starting prop forward. In the post-season he was selected to make his debut for the New Zealand national team in the same position in a game against Australia in Newcastle. At the 2011 Four Nations tournament held shortly after, Packer played one match for New Zealand against England.
In June 2013, at Suncorp Stadium, Brisbane, Queensland, Packer was caught on camera urinating on field prior to kick off. Packer then played out the match for the Warriors against the Brisbane Broncos. The incident, dubbed "completely unacceptable" by the NRL, resulted in a $15,000 fine. After the game he tweeted "Good win when u gotta go u gotta go lol".
On 2 October 2013, Packer signed a 4-year contract with the Newcastle Knights starting in 2014. In January 2014, Packer's contract was terminated before he could play for the Knights due to a serious assault committed by Packer at the end of November 2013 and impending criminal proceedings.
On 29 November 2013, Packer appeared before court in Sydney, New South Wales, charged with assaulting a 22-year-old man, fracturing his eye socket. Packer punched the man in the face, causing him to fall and hit his head on the ground. Packer then punched the man several times as he lay on the ground and stomped on his head, leaving the victim with two fractured facial bones. On 6 January 2014, Packer was sentenced to two years in jail after pleading guilty to assault occasioning actual bodily harm. Magistrate Greg Grogan labeled Packer's actions as "cowardly and deplorable".
In April 2014 the non-parole period of Packer's two-year sentence was reduced on appeal from two years to one. Having had a criminal conviction, there had been fears that he would be deported back to New Zealand, but on 16 December he was granted permission to remain in Australia, thus clearing the way for him to resume his first grade career with the St. George Illawarra Dragons.
In May 2015, Packer returned to rugby league with the Dragons' second-tier team the Illawarra Cutters in the New South Wales Cup. On 27 September 2015, he was named at prop in the 2015 New South Wales Cup Team of the Year. The Dragons had been assisting Packer's rehabilitation within the rugby league community throughout 2015 and in October of that year, after the NRL gave formal approval for him to play in the NRL once more, he was able to sign a 2-year contract with them, starting in 2016.