Question: I have given notice at my current company and have admittedly done a poor job of organizing all of my contacts, so sending out an email to all of them would be tedious to say the least. Everyone internally knows of my departure, it is the people outside of my organization that I am concerned about.

Would it be generally appropriate to include the fact that I will no longer be with my current company in my email signature? I feel that giving this notice to everyone will cause a lot less headaches for them down the road.

Is it even my responsibility to inform them that I am leaving? (I have no problems doing so)

I also plan on setting up my out-of-office reply to include this information once I leave to handle any overlap between incoming emails and my company shutting down my accounts. Is this appropriate as well?

Note: Everyone internally knows of my departure.

Answer: 
Talk to your company/supervisor. They may/should have a plan for what to do with your email account and phone number. They should even have a plan for how to inform customers and vendors.

This is one of the items that takes place during the transition period. Though I have never heard of a company that wanted to use am email signature to spread that information.
Issues of professionalism or appropriateness aside, using your email signature to announce your departure is likely to be *ineffective*. The vast majority of email clients de-emphasize or hide signatures by default, so most of the people you're trying to contact with this method will never see it.