I am writing my own stories and will self publish them on my own website.
One of the stories has been inspired by a short story by Ray Bradbury, indeed the central tenet forms the basis of my story.
One thing I need to know is about Copyright, does anyone know who I will need to approach regarding writing my story?
This is very much a work in progress, indeed it has been ongoing for years, but I would like to get rolling on this as soon as I can.

A lot of writers have used similar ideas over the years. Is this concept absolutely unique to this one story?

You would need to approach Don Congdon Associates (Google them for the contact details) if you are serious about obtaining permissions - but I agree with dandelion:

If your story just uses a similar premise - space travel, time travel, small town, strange psychological happening - go ahead and write your own thing.

And if your story is so close to Bradbury's as to potentially infringe copyright, why even bother writing it? Most people would probably prefer to read the original!

A couple of other things...

Copyright only applies at the point of publication, so if you wrote the story for your own amusement and never published it, you can do anything you like.

If you're publishing on a personal, non-commercial website, and you're giving the story away for free, I think you'd be fairly safe in labelling the story as "fan fiction" and being upfront about it being an homage to Ray. That doesn't protect you legally (copyright infringement is copyright infringement), but it will minimise the risk that anyone will worry about you as a threat to Ray's copyright.