To get a technician level job, you need to have a Bachelor of science (BSc) majoring in chemistry, biochemistry, biotechnology, medical laboratory science or molecular biology. However, most employers nowadays prefer a Masters in science (MSc) in any of these areas.

Those who want to do DNA analysis need to have a BSc in molecular biology as a minimum. It is increasingly common for those graduates to then do a postgraduate diploma or Masters in forensic science.
  
Most successful applicants (about 70%) for permanent positions as forensic scientists have a Masters in forensic science. Some may have a PhD. 

A tertiary entrance qualification is required to enter further training. An A or B Bursary, or NCEA equivalent, is preferred. Useful subjects include maths with statistics and/or calculus, biology, chemistry and English.

Many practical forensic science skills are gained on the job. Once a person is employed they are usually trained in the specific scientific techniques that are involved in their job.

Useful experience includes laboratory work, and research or work in the fields of medicine or chemistry.

Previous experience doing casual work for Environmental Science & Research (ESR) might also be helpful.

Check the ESR website for information on summer studentships for university students.

ESR website