It would be really great to be able to do spellchecking right inside WordPress. Specifically, it would be great to be able to check the entire site in one shot. That would require something working from inside, ie, a plugin.
I looked at a lot of spellchecking plugins, and I didn't find one that was acceptable. It's not that they don't actually work, but every one seems to cause non-fatal errors, and I don't want faulty code running on the site. But I could still use some sort of tool, as this website is cobbled together from a huge mass of forum posts that were not all written with the greatest care.
Finally, I tried a browser extension: Grammarly. Grammarly adds spell checking to any html input field. This adds spellchecking to the WordPress editor. The downside is that it only checks one page at a time, and that page has to be open in the editor. That's not that bad, you'd have to open the page to fix anything anyway.