Go to your private message inbox.
Find the messages you would like to delete.
Look to the right of the message and you will see an empty square. Click that square to select the message for deletion.
Scroll to the bottom of the page and you will see a large button that reads "Selected Messages (*)" The * represents the number of messages you have selected to delete.
Choose Delete from the drop-down that appears after you have clicked the "Selected Messages" button.
Click "Proceed" to delete those messages.