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.
I would be careful in doing this. Check the terms and conditions of your employment contract and the laws in your country, to make sure you are not in breach.

I know someone who did this. His contract was subject to a clause which prevented him from working with any of the company's clients within a certain period of leaving that job.

He sent an email to all his clients to inform them that he was leaving (he had good reasons for doing so which I won't go into here). The company took him to court and obtained an injunction against him. They were able to successfully argue that by informing the clients that he was leaving, those clients (who may have valued their relationship with him more so than the company) would be able to easily locate him at the next company he worked at, and switch their business.