Check out
www.loot.com, which is the English version of segundomano for places to rent. Not sure how much you think cheap is though, cos generally London isn't.
With regards to location, you really want to be on the underground (Piccadilly line) or a local bus route. The closest big town is Hounslow/Hatton Cross both of which have bus/tube links with the airport. Feltham is a bit more desirable, also has a bus link and is on the main railway network too. A little further afield but on the same line is Acton and Chiswick, but the further you travel the more you pay on the underground.
Hope that helps, Graham.