Deleting messages on Facebook is not as easy as deleting your  email, as Facebook intends to keep your conversation history in Messages  and Facebook Chat intact.  
To delete messages on Facebook, you will need to open the individual  message, go to options and start selecting the message you want to  delete. If you have many messages, the process can be very  time-consuming. Otherwise, you can simply remove the messages from the  message list, but doing this will not remove the message permanently.  Instead it will be archived, and will reappear when the person sends you  a new message. 
 Here’s where we bring in some help i.e. ‘
Facebook Fast Delete Messages‘, an extension for your Chrome browser that will allow you to delete Facebook messages with a single click. 
deleting facebook messages (Slow) 
You can open your Facebook messages and you can select to send your  messages to ‘Archive’, one by one. However, the message you delete will  not be removed permanently; it will reappear when the person sends you  new messages. 
To delete individual messages, you need to open each message first. Do this by clicking on the 
Actions button and select 
Delete Messages. 
You will see all your messages with this friend, with checkboxes on  the side of each message. Select the messages you want to delete by  ‘checking’ the box next to the message, or simply click on the 
Delete all button. 
It’s tedious, isn’t it?  
Delete all facebook messages at once
To take control of your Facebook messages, go to the 
Facebook Fast Delete Messages extension page with your Chrome browser and click the 
Add to Chrome button to install. 
Once installed, the extension is activated automatically. Now to  delete your Facebook messages with only one click without having to open  the individual conversation pages open your Facebook Message page. You  will see two extra buttons, one at the top bar and another highlighted  in red. Click these buttons to permanently delete your messages and  conversations. 
Conclusion
With this chrome extension, you don’t have to go through the long  process of message deleting, forever clicking to delete your Facebook  messages. Get the extension installed onto your Chrome browser, and you  can now delete messages with a single click!