Step 0: Make sure your starcraft client is up to date and fully patched
Step 1: Open up the Starcraft 2 map editor
Step 2: File, Open, select the campaign tab, choose "ZLab01.SC2map"
Step 3: Press the "Test Document" button (looks like a green SC2 logo)
This will dump you into the very first mission. You'll miss the very first cinematic (spoiler alert: this expansion is about zergs), but you can go back and watch it from the archive later. Just save your progress and you'll be able to play through the entire game, not including multiplayer of course.