Reference: Fataawa li Ahlil-Mayyit – Question 3, Page 8
Question: What is the correct manner narrated from Mustafaa (sal-Allaahu ‘alayhi wa sallam) in washing the deceased?
Response: The legislated manner in washing the deceased is that someone should [first] wash his private area under a covering [to protect his ‘awrah], thereafter, it is legislated to wash the rest of the body.
So, you begin with those parts of the body to be washed in wudhoo, hence, you make wudhoo for him, except that you must not insert any water in his mouth or nose, rather you should moisten a piece of cloth and clean his nose and mouth with it.
Then you should wash the rest of the body, starting with the chest.