I will assume you are working on a fresh, installation of Alpine standard. Otherwise see these instructions.
Enable Community Repository
We need the Community Repository available for access to the i3 packages. Instructions are available on the Repositories page.
Ensure you have the required drivers.
These are the drivers I needed but the ones you need might differ.
Install some fonts. If you start i3 without fonts it will be unusable.
Install dbus and set it up.
Install X11 and set it up.
Install i3 etc
Install i3, the i3 status bar and a terminal.
Set up a non-root account
Set up a non-root account. (Call this whatever you like.)
Now you can start i3 manually.
Note: The default configuration is minimal and most menu options won't work unless you customize and/or install additional packages. More information is available in the i3 user guide.
Starting i3 on login
To have i3 start automatically on login, you should set up xinit. There are different ways to do this. The arch wiki has good, transferrable information on this.