By default, WordPress is very, very careful about who can delete files in your site's Media Library.
Only users in the "Administrator" role are able to delete images and files in your Media Library. Anyone who is in the Subscriber, Contributor, Author and Editor roles is not allowed to delete images.
However, on some sites you may want to allow other users to clean your Media Library. Perhaps you have private files, or just have a cluttered Media Library with too many files.
The solution to deleting files is to install the Capability Manager Enhanced plugin.
Once both those plugins are installed, go to Permissions > Role Capabilties in your WordPress admin area.
In the right sidebar, check the "Media" box under "Force Type-Specific Capabilities". This will allow you control the permissions for the Media Library,
In the top-right corner of the screen, load the user role that you want to customize permissions for. In this image below, I've chosen the "Editor" role (click here to read about Editors capabilities).
In the center of the screen, you can now set your permissions. If you want to allow people in this role to delete anything in your Media Library, check all the boxes in the "Media" row.
Click "Save Changes" at the bottom of the screen.
It's as simple as that. Now whenever a user in the "Editor" role goes to the Media Library, they'll see a "Delete permanently" option for any file they see: