Applying for a Letter of Probate without a solicitor (Australia). Save at least $1000

Most people go straight to a legal firm or trustee company, probably whoever is holding the will of the deceased. This is not necessary in most cases. You can spend $1500 - $2000 on an average estate (not including Supreme Court costs, see below).
Request the will, if it is held by a solicitor. The funeral director organises the death certificate.
With these two documents, you are ready to do-it-yourself.
You need:

• Original Will
• Death Certificate
• A computer and printer
• Application forms (filled in by typing or clearly printed)
• Advertisement (see below)
• DIY Probate kit suitable for your state/territory (recommended but not essential)

If you live in the state of Victoria, Australia, paste this in your browser:

https://www.supremecourt.vic.gov.au/forms-fees-and-services/wills-and-probate/applying-for-a-grant-of-probate-or-administration

Other states or territories will have similar sites.

Supreme Court of Victoria website is clear to follow, set out step-by-step. For extra help and tips, you can buy a D.I.Y Probate Kit online that includes stationery and application form templates according to a wide variety of cases - e.g. original executors, substituted (in the event of death) executors, multiple executors and beneficiaries.
The Supreme Court gives you a link to LibreOffice (used for entering in your application for probate that you simply print out and take to the Probate Office in Melbourne).

Costs (in Victoria, as of June 2018):

Advertising your intention to apply for probate through the Supreme Court website) $45.50
Application fee (paid on the day you go to the Probate Office)  $320

Optional D.I.Y. (I used Legal Kits of Victoria) guide $99 + $15 postage (money well spent)

In my situation I found the only tricky part was finding out the full names and addresses of witnesses to the will (written nearly 40 years ago, it there were two indistinct signatures). However, my father's solicitors had this information on file and emailed me this (at no cost).
The staff at the Melbourne Probate Office (Level 2, 436 Lonsdale Street) were friendly, helpful and not intimidating to a legal-newbie like myself. Note, the office is only open on weekdays, 8:30 a.m. to 4.00 p.m.