Hi Alex
You can save a wiki to your Dropbox and generate a public URL for it (they call it "sharing a link"). The URL will be of the form:
(Where the dashes stand for other encoded characters that I have removed)
You need to transform the URL to this form before the wiki will view correctly:
Depending on the way that your domains are managed, you should be able to set up a redirect to that URL.
The easiest way to get your own private instance of TiddlyWiki5 running on your own server is to use the old UploadPlugin that was originally developed for TiddlyWiki Classic. I've recently updated the docs for 5.0.7 here:
You'll probably be best off waiting for the release of 5.0.7 before trying it out (5.0.7 will be out in the next couple of days)
Best wishes
Jeremy