You can delete private messages from someone by logging in to reddit.com or old.reddit.com on your computer’s web browser. (Sorry, but you can’t delete private messages from the Reddit app on your phone.)
Once you’re logged in, go to the private message you’d like to delete and click the delete link under the message.
If you have more questions that aren’t covered in these FAQs, check out r/help. Someone may have had the same question, or you can ask a new one and a Reddit administrator or one of your fellow redditors will help you out.