Check that only one version of your site is browsable.
Run a website crawl.
Check for indexation issues.
Check that the site ranks for their brand name.
Perform on-page SEO checks for the main pages of site.
Check on-page issues in the crawl report.
Check for duplicate and thin content.
Check that the site (and individual pages) load fast.
Check for structured data errors.
Analyze organic search traffic.
Check that keyword rankings are headed in the correct direction.
Find low-hanging keyword ranking opportunities.
Analyze the site's backlinks.
Find and fix broken links to-and-from the site.
Find content gaps.
Conduct a full content audit.
A simple example of a prior website audit.