What is a 'British' accent? You've got either Scottish (Doric from the North-east, a more posh accent in edinburgh, and yet another accent in Glasgow), English (Cockney, upper-class, etc. -- too many variations to go through), or Welsh (a bit more uniform, as Wales is smaller) accents. Britain is not a country, but an island made up of three distinct cultural areas.
To be honest, the best way to learn an accent is immersion - I've picked up a NE Scottish accent in around a month of communication. Otherwise, try and seek out oral examples of the accent (not Hollywood movies; they don't know their accents) and try to listen to as diverse a range of words and phrases spoken as possible. I'd really advise for immersion, though. You'll pick up something odd trying to do it any other way.