How do I remove the previous App Store user from my computer?
The App Store app uses spotlight to find all the apps on your Mac that were downloaded from the store. You need to run a search and delete all those apps from the Mac (your personal data and settings do not get deleted) and re-download the app. Remember, you could have more than one copy of an app, so keep checking until they all are deleted.
Step one of course, it to sign out from the other account if you haven't done that yet. When the App Store App is running, choose the Store -&gt; Sign Out menu item.
The details on how the App Store app actually knows if an app was downloaded is buried inside the application bundle as descrtibed on this quesiton about avoiding updates to apps entirely.

How can I opt out of updates for one app when using the App Store?