The best way to access your Zotero library from multiple computers is to sync your libraries automatically via the Zotero server. If, for whatever reason, you cannot use Zotero's built in sync features consider the options below.
If you are unable to use Zotero syncing, you can also store only your Zotero data directory on an external drive and use this to move your Zotero data between computers. You can set your Zotero installation on each computer to point to the external drive in the Advanced pane of Zotero preferences.
Another option is to run a copy of Zotero directly from a portable drive. See here for more information.