Many Americans melt at the sound of a British accent. But, as Dick Van Dyke demonstrated in Mary Poppins, a bad imitation of a British accent is an unpleasant thing. For actors who might need to be able to pull off the accent, and others who just want to, the key is immersion and imitation. The upper-class British accent is often referred to as BBC English or received pronunciation. Several programs are available for purchase to learn it, but most just recommend listening and imitation. If you don't have a good ear for accents, ask a friend to critique.
Watch British movies. This gets you used to the way British people pronounce letters and words differently. For example, Americans use hard "r"s while British people pronounce "r" in a way that sounds like "ah." Immerse yourself in British English and try repeating phrases and passages from movies you watch.
Study speech patterns of actors like Hugh Laurie and Minnie Driver who, though British, pull off a convincing American accent. Note the difference in the way they pronounce words when playing American characters versus when playing British characters or during interviews. Pay attention especially to the differences in vowels and use of the letter "r."
Make use of websites such as the Blogspot Learning British UK Accent for specific instructions on vowel and consonant sounds that are significantly different in American English vs. British English. Practice the vowel sounds and phrases given.