It's a passive voice noun clause with a reduced participle clause. The full version would be: Whatever you don't like to be done to yourself, don't do to others. Whatever just means 'the thing(s) that'. Don't do the thing(s) that you don't like (to be) done to yourself to others. Don't do to others that which you don't like (to be) done to yourself. For active voice: Whatever you don't like people to do to you, don't do to others.

"Done to yourself" is an adjective phrase that modifies the object, which is "whatever". You'll see it more clearly using "something": "You don't like something done to yourself".

The usual.way we phrase that is: Do to others what you would have them to to you.

I have same problem