Now if you like to Delete All Tweets then there are few websites which can help you delete old tweets. You simply need to authorize those website on twitter & then you're good to go & delete all your tweets.
Few of those websites which can help you to delete tweets are:
1. Tweet Delete: http://goo.gl/bXvTsV
2. Delete All My Tweets: http://goo.gl/ukYUXA
3. Tweet Eraser: http://goo.gl/3YicqN
4. Tweet Deleter: http://goo.gl/2mB0vZ
5. You can also delete all your Tweets on Android via DLTTR: http://goo.gl/Difd4E