Your solution is two-part:
- Backing up tweets
- Replaying them
For number one, it's easiest to just have an RSS reader subscribe to your twitter feed and save that somewhere. The benefit of using RSS is that it serializes it in a format that lots of other software can read/write (it's all XML). Unfortunately, Twitter doesn't offer RSS directly from their site, so you have to use something like TwitRSS.me
Alternatively, you can request a backup of your tweets from your twitter account.
For number two, it's a little bit more difficult. If you have any development experience, you could use Twitter's API to post all your old tweets.