We have 2 types of e-mails at the social business network company I current work with:
What I'm doing first when the developer tells me she is finished, I send it to the Litmus.com service to get a first quick impression what happens.
Litmus.com or EmailOnAcid.com, both provide you with screenshots from different clients for one e-mail you send to them.
Use the watering can approach to get quick a broad idea how your e-mail behaves on different platforms and clients - shows how robust your code is.
Then I run the HTML syntax checker, I prefer Firefox browser with a plugin and just the HTML code of the e-mail from a saved file:
Algorithm: HTML Tidy
Usual I got told before by the developer where she already made some compromises, or where some effort did not solve some ugliness, yet.
Since she anyways develops by verifying with Litmus.com I do it again to get the same final picture. Depending on the results I verify some unbelievable results with real devices.
If the content is not static and involves dynamic input I select some minimal and some maximal input values and resend the e-mail with these settings.
Our self developed templating system allows different files with made up data to feed the variables of the e-mail, and displaying the e-mail in a browser or send it to a real e-mail recipient.
To get the same result, as your real recipients will get, use the same system/method to send your test e-mail as you would end up with the final live one, do not use the forwarding function of your e-mail client to show the e-mail to another person, when you want to be sure the e-mail looks the same as on your client! Forwarding is likely changing the content of the e-mail, when forwarded.
There is a use case when your e-mails purpose is to get forwarded.
More details: https://litmus.com/blog/when-forwarded-emails-break
After that I select some mobile devices; one iOS, one Android, to interact with the e-mail, click it, use horizontal/vertical switching, load pictures, when previous they are not loaded.
Checking what it looks like without loaded pictures is an important point, there are many clients who prevent loading pictures per default.