You can use the sync feature of Zotero to use your personal Zotero library on multiple computers, including a library computer.
- Go into Zotero preferences, select the "sync" tab, and add your zotero.org login and password.
- You can either select "sync automatically" or periodically click the sync icon
on top right in the Zotero application.
- Remember to remove your credentials from the preferences when you are done.