There are two ways to handle the deleting of unwanted email within Apple Mail. You can move mail to a Trash folder until it is time to permanently remove the messages, or you can keep them in your Inbox until they are to be removed completely. Deleted messages will remain in the Trash until they are permanently erased, as specified in Mail preferences. If you want to empty the trash early, you can Right click or [Control] click on the trash icon and select "Erase Deleted Messages."

In Mail preferences, under the Accounts tab, you can specify for each mail account to not "Move deleted messages to the Trash mailbox" by un-checking the checkbox. Messages remain in their original location after you delete them, and are shown dimmed or are hidden. To show or hide deleted messages, select "Show Deleted Messages" or "Hide Deleted Messages" from the View menu.

You will probably want to select the "Hide Deleted Messages" options to get rid off those grayed out messages.