The correct order is most important. It is your claim to file and usually it must be done formally within 60 days of the loss, or have a written extension of time to file. However, before filing your claim you must know what you are entitled to and some of that you appear to know with your research of market value.
First, here are the rules for determining automobile actual cash value:
See Full article in the book, "Policy Ensurance"
Now that you know the rules to determine the amount of your claim, you need to document your research on the value of your automobile, i.e., actual newspaper ads, dealer estimates of value, etc.
Next comes filing a formal claim. That is usually done by filing a "Proof of Loss," form with the insurance company.
After filing your claim if it is contested by the company and they seek "appraisal," a form of arbitration, the disinterested appraiser that you select only needs to be someone who understands your claim and the rules for determining the amount of your loss and claim. It could be an attorney, adjuster, accountant, etc., anyone who understands the process too would help.
If you are interested, the claim and appraisal forms and procedures are all contained in the book, "Policy Ensurance," available in libraries and major book stores: http://disasterprepared.net/contents.html
Best of luck with your claim.