Permanently deleting conversations from your Inbox requires two steps.
First, click on the "..." dropdown in the top right corner of the conversation and select "Move to Trash" from the menu. You will then be asked to confirm this action.
Next, in the header area you now have an option to "Delete forever". You will also have to confirm this action.
Note that deleting a conversation forever is indeed permanent and the conversation cannot be recovered. Only admins have the ability to delete conversations.
Bulk Deleting Conversations
To bulk delete conversations permanently, select the tickets you want to delete from your Trash folder and then click the trash can button that appears in the center of the page.