Twitter has been slowly rolling out its archive facility to users on request. Even once you've ganed access to the archive, it's a static document. But using a handy script in conjunction with Google Drive lets you keep it up-to-date with all the tweets made after you've requested it.
Martin Hawksey goes into detail on how to use Google Drive and Google Apps Script to achieve the result. The embedded video covers the basics well; hit his post for the detailed walkthrough.