An SEO audit must be fair and objective, so always be supported by figures. You can use a lot of tools for this. The list below is certainly not complete, but will get you started:
Crawler that checks and analyzes your onsite SEO. For PC, Mac and Linux.
Deep Crawl Alternative crawler that works from the cloud, so no installation required.
SEO Quake For on-page SEO audits, control of internal and external links and more.
Web Developer Plenty of useful tools under one roof. Extension for Chrome, Firefox and Opera.
Google Structured Data Testing Tool Check microdata from pages.
Google PageSpeed Insights Test the loading time of web pages. A good second opinion is Pingdom
Check externally for duplicate content. Free alternatives that one could consult are: Plagium and Duplicate Checker
Find duplicate content, broken links and more within the site. As for broken left, Ahrefs is also an option.
Barracuda Tool that analyzes how Google algorithm changes have affected a site.
SEMrush For those who need keywords and want to know what the competitor scores on. An alternative could be SpyFu although it seems to have less organic search engine optimization data for the Dutch market than SEMrush. For campaigns in Google.nl, I personally prefer SEMrush for that reason.
Google Search Console Various tools to check the optimization of a site, including backlinks and keywords.