Depends how far back you're looking. Some papers - Telegraph, Times, Standard - have online searchable archives at their regular websites. But they're likely to go back no further than the websites themselves - a few years at most. To go back further you have to start looking offline - most libraries hold a newspaper archive - the local rag and the Times, in the case of my local. And the Guardian has recently opened an archive centre near its offices in central London. At the same time they launched a website with some articles from the last 100 years - which may have what you're looking for:
http://www.guardiancentury.co.uk/