Comandante has made a very good point here. There is no such thing as 'the British accent'. For someone actually from Britain, this term is completely meaningless. There are many different accents used across different regions of England, and there are also many different Welsh and Scottish accents. The accent of a person from London, for example, may be very different from that of a person from Manchester or Glasgow. In fact, to an outsider, a London accent might sound closer to an Australian accent than to a Scottish one.

Listen and speak to British people. Be ware of what accent you want to learn. Do you want to learn London accent? or which one exactly? I come to realize that there are about 8 major accents and about 12 minor accent in UK so it is your decision.

listen from britishcouncil.org, bbc.com/new, bbc learn English and talk to British people on skype..:)

correction~ Be Aware of

Hi Ale. Lessons in conversation I feel would help the most in helping you acquire the British accent. Other than the obvious best solution of going to live in England, you could look through the teachers on italki, especially checking out the teachers from England, and have a few trial sessions.