Embarking on an acting career, in the early 1990s she was most well known for her role as Mick Johnson's wife Josie in the Channel 4 soap Brookside. She then undertook a degree in teaching at Goldsmiths College, and after graduating in 1996 taught drama in London.
Arriving back in Cardiff with her three-month-old son, she gained an audition for BBC drama Casualty. Three weeks later she started work as Sister Tess Bateman, first appearing on screen from September 2003. In November 2006 Martina Laird who played the role of Comfort left Casualty, which then meant that Suzanne was the second longest serving cast member.
Packer is divorced from American actor Jesse Newman and in 2006 lived in Cardiff with their son Paris.
Packer appeared with her brother Colin on the first series of Pointless Celebrities on 6 July 2011. They were eliminated in the first round.
On August 22, 2015, she quit Casualty after playing the character of Tess for nearly 12 years to become a member of the cast of Stella and Doctors.
On 19 March 2017, Packer appeared in the ITV crime drama Vera as ranger Sophia in the episode "Natural Selection".
On 18 March 2018, Packer played a minor role in an episode of Hold the Sunset.