Install the plugin in the usual way. Go to Appearance > Widgets and drag the Mixtape Machine into your sidebar. Now head over to SoundCloud, and open the playlist, track or user stream you want to include on your site. Click the Share button, then the Embed tab. Select and copy the standard Embed Code (not the WordPress one) and paste into the widget controller on your WordPress admin screen. Pick a background color, save and you’re good to go.
Consider using a plugin like Advanced Ajax Page Loader on your site, to allow users to navigate between pages without interrupting the flow of the music. I’m using it here.
Alternatively go for a single-page theme, or one with AJAX page loading built in.
Autoplay if you absolutely must
You can still customise the native SoundCloud player with all its built-in settings (these will be displayed when users click the cloud icon on the tracklist), including the autoplay option, so visitors to your site can brighten up their colleagues’ workdays with a few unexpected blasts of your tunes from their speakers.