I measure backlinks with Ahrefs' free Backlink Checker tool.
It gives a nice overview of your estimated Domain Rating, number of backlinks, and number of referring domains.
This tool also nicely breaks down how many of your backlinks are "dofollow" links that give you more points towards your page's rank.
Ahrefs' Website Authority Checker.
It gives you a measure of your website's authority based on its backlink profile. Search engines will typically rank you better if your website has more authority.
I typically run my websites through a tool such as Seobility's SEO check.
Speed and Performance
I regularly measure my pagespeeds through Google's own PageSpeed Insights tool.
Spelling mistakes can be embarrassing.
Typosaurus is my current favorite spellchecker. Paste in multiple URLs and they tell you if any words aren't spelled correctly.
I typically use Grammarly to help avoid any blatant grammar mistakes. They have a nice Chrome extension, but also a simple web interface.
You can easily check your content to make sure it's clear and concise.
I try and paste all my new content into the Hemingway Editor to make sure my writing is clear and easy to read.