Study the Queen's English dialect. Consult reference books and news articles that discuss the various attributes of the accent, such as how "r" sounds can be placed in words that don't contain an "r" (i.e. "really" sounds like "rairly" and "off" sounds like "orf"). Practice saying such words to get a feel for how the words roll off your tongue.
Watch interviews and other videos featuring British royals such as Queen Elizabeth II. Listen intently to how the royal person speaks, how she forms her sentences and how she pronounces certain words. Pause the videos as needed and mimic the person and repeat what she says so that you can sound like her.
Use the royal "one." Many British royals and other speakers of the Queen's English use the word "one" instead of the words "me," "I" or "you" to sound more formal. For example, instead of saying "You're upset, aren't you?," someone speaking the Queen's English would say "One's upset, isn't one?"
Utilize a more classy vocabulary of words. Replace "common" or unrefined words with a more tasteful alternative, such as using the word "cad" instead of "jerk," the word "dame" instead of woman and the word "fiendish" instead of "sneaky" or "tricky." Never use the phrases "By God!" or "Oh my God!" in front of ladies or clergymen. Instead use a phrase such as "By Jove!"
Practice speaking the Queen's English as much as possible. The more often you speak in proper Queen's English the more natural your accent will sound and the easier it will be to speak. Refer back to any of the aforementioned learning tools to aid yourself.