Most people who want to speak English with a British accent are immigrants to Britain from non-English-speaking countries. They'll often watch the BBC and try to pick up the subtle nuances and quirks of the language as they hear it. If you're an American, you can use many of the same tools as these folks to develop a well-rounded British accent.
Choose a national accent to develop. Great Britain contains the nations of Wales, Scotland, England and Northern Ireland, and each one has a different predominant accent.
Narrow your accent choice down to a specific regional dialect. Received Pronunciation or "Queen's English" is the standard high-brow dialect in England, but every geographical region has its own flavor.
Listen to clips of British people talking in your selected accent. The BBC's website's a good place to look first for digital recordings of specific geographical accents like the scouse and cockney variations.
Spend time practicing the accent. Instead of focusing on quantity, improve the quality of your practice sessions by recording yourself speaking a line or two of text and then playing it back.
Analyze your mistakes. When you slip up and lose the accent while you're talking, make a note of the word you missed. This allows you to focus on problem words and phrases in later practice sessions.
Travel to Britain and hang out in a particular town. The more time you spend in one place, the easier it becomes for you to adopt its accent.
