The best way to access your Zotero library from multiple computers is to use Zotero Sync. Zotero Sync will automatically sync your library data using the Zotero servers. Attachment files can be synced using the Zotero servers or using a third-party WebDAV servers.

You absolutely must not store your Zotero database in a cloud-syncing folder (e.g., Dropbox, Google Drive, OneDrive); this will lead to data corruption. You can configure Zotero to sync your attachment files using these services while using Zotero Sync to sync your library data.

If, for whatever reason, you cannot use Zotero's built-in sync features, you can also store your Zotero data directory on an external hard 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.