Instructions and status as of August, 2016.
Modern Firefox or Chrome (IE will work, but badly)
A WebSocket proxy
websockify: https://github.com/kanaka/websockify works great.
Note that a patch to remove this requirement has been submitted to the Spice project but not yet been accepted. Refer to this email: https://lists.freedesktop.org/archives/spice-devel/2016-June/030552.html
A spice server
A web server
With firefox, you can just open file:///your-path-to-spice.html-here
With Chrome, you have to set a secret config flag to do that, or serve the files from a web server.
Start the spice server
Start websockify; my command line looks like this: ./websockify 5959 localhost:5900
Fire up spice.html, set host + port + password, and click start
The TODO file should be a fairly comprehensive list of tasks required to make this client more fully functional.