You should first collect tips by googling different articles- "how to do a British accent". Then watch lots of British movies and TV. As far as good examples go, I recommend Atonement, Pride and Prejudice, The King's Speech, Doctor Who, Downton Abbey, Sherlock Holmes (the old '80s TV show), A Christmas Carol, The Lion the Witch and the Wardrobe, and Lord of the Rings (for the most part) to name a few. The best way to speak with a British accent is to listen to one. Watch the actors' lips, watch their teeth, watch their tongues and their jaws. That will help you. Also, you can trust a Scottish actor to imitate a Brit properly for the most part but many times, Americans botch the accent. Not to say an American can't do it- but be careful who you trust. After a while you'll be able to tell what is authentic and what isn't. Keep in mind there are many British accents- cockney, posh, Londoner, Yorkshire- learn the differences and know what you want. Listen to everything you can, practice different words. You'll have it down in no time.
You need to find a dialect coach. Actors use them all the time. You will also want to discuss which style you wish to imitate. There are several different ones in and around London itself, and even more as you move outward from there.