I am trying to limit it so you can only delete the files you uploaded yourself. Also, make it so you can only delete files not folders.
You can achieve it using designer workflow:
- Upload a file.
- Trigger a workflow, which grants Delete permission to Created By.
- User has enough permission to delete the file he uploaded.
How to set item level permission using designer workflow: http://spcycle.blogspot.sg/2012/01/how-to-create-workflow-to-change-item.html