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.

One option is to run a copy of Firefox directly from a portable drive, which gives you access to your Firefox settings, such as bookmarks and history, in addition to your Zotero data. See Portable Firefox for more information.

If you use Zotero only on your own computers and don't need to keep your Firefox profile in sync, there's no need to use Portable Firefox. Simply copy your Zotero data directory to the portable drive and set Zotero on each computer to use the same custom data directory on the drive. You can set a custom data directory in the Advanced tab of the Zotero preferences.