If you have a big Twitter account and you want to completely delete all your old tweets in one go, here is how to do it:

Tutorial: How to delete all your Tweets for free (code and userscript in comments)
byu/lucahammer inTwitter

1. Log into your Twitter account.
2. Export your Twitter data archive.
   How to: https://twitter.com/settings/download_your_data
   (It may take 1-2 days depending on the volume of your data)
3. Unzip the file after downloading and locate a file called tweet-headers.js.
4. Copy script from https://gist.github.com/lucahammer/a4d1e957ec9e061e3cccafcbed599e16#file-tweetxer-js
5. Back on your Twitter profile page, hit F12, then select the Console tab.
6. Type “allow pasting” and hit enter.
7. Then paste the script from Step 4 and hit enter.
8. Select tweet-headers.js from Step 3.
9. Watch all your tweets being deleted one by one.