The best way to learn how to speak British English is to repeatedly listen to British people speak. Turn on the British Broadcasting Channel (BBC), and listen to how the newscasters and actors pronounce various words and how they put them together in sentences. The British use a more formal manner of speaking than many Americans are used to. Also try downloading audio files from the Internet of people speaking British English. For easy access and to listen to them while you are in the car or shower, load them onto an MP3 player. When you hear the words pronounced on the audio, you will have a better chance to duplicate the sounds than you would if you were just reading about how to learn British English.
Many Americans view British English as being sophisticated because the accent provides a crisp, clean sound. Do not leave -ing's off words such as coming and going to make them comin' and goin'. Do not run words together. Instead of saying "Jeet jet," say slowly and clearly, "Did you eat yet." Place extra emphasis on a "T" if it is at the end of a word, such as "let."
If you are going to speak British English and be convincing, you must not only speak with a British accent, but also use the correct slang terms. Some examples of words that Americans and Brits have different terms for are: elevator/lift, restroom/loo, gas station/petrol, money/quid, french fries/chips, potato chips/crisps, kiss/snog, friend/chum. It would be wise to purchase a British slang dictionary to use as a reference.
As you practice your British English, record yourself. Listen to the recording and ask others to listen to it. Other people may hear you speaking differently than you hear yourself. Do not overemphasize the British accent.
Practice your accent by talking with British natives. They will quickly catch any errors. If no British natives are available, try speaking with someone who has already learned British English and is proficient at it. Learning to speak British English well may require a lot of time and effort to perfect.