If you want to programmatically interact with Dropbox using PHP, I recommend using the official PHP Core SDK:
https://www.dropbox.com/developers/core/sdks/phphttps://www.dropbox.com/developers/core/start/phphttp://dropbox.github.io/dropbox-sdk-php/api-docs/v1.1.x/
How you zip up the file isn't related to the Dropbox API, but to upload large files you can use:
http://dropbox.github.io/dropbox-sdk-php/api-docs/v1.1.x/class-Dropbox.Client.html#_uploadFileChunked
