How do you write an expression of interest for employment in a specific position in private?

An EOI is a covering letter introducing yourself, saying why you would like to do the job. No specific format is required. The selection panel will not be expecting anything too long, it is just literally expressing interest in the job and is more personal than a Resume.

The letter should provide only a few highlights from your resume that are specifically suited to the position. The resume should be attached and it's not necessary to repeat much from it. The purpose of the letter of interest is to give the employer a reason to contact you for an appointment for an interview. At the interview you can expand on your training and experience and how it suits the position.

It is also a good idea to research the position on offer, particularly for Government positions, and read any relevant policies and guidelines. A candidate who can write or speak knowledgeably about the company or organization will have an advantage over those who have not done their homework.