Listen to the BBC World Service. Received pronunciation was adopted in 1922 by Lord Reith, the general manager of the BBC, as the broadcasting standard. Though regional accents are now much more acceptable, many presenters still use RP. This is one variation of what may be termed "posh". Other people associate a posh accent with members of the royal family. This is not quite the same as RP. For instance, members of the aristocracy have a tendency to pronounce "house" as "hice." Not very many people in Britain today speak like this.
Pronounce your words properly. Do not over-articulate, however, as this will sound false. Use short vowel sounds. Do not cut off the ends of words in a lazy manner, particularly consonants. These should be quite clipped. Be aware that there are many English accents (not to mention Welsh and Scottish). There are also subtle variations even within the regions: for instance there is a strong or broad Liverpool accent (often called "Scouse"), whereas there is a much more refined Liverpool accent, which is milder.
Watch lots of British films, learning the subtleties and the various regional accents and dialects. Listen to audio books narrated by British actors. Visit a voice coach and ask for lessons. Soften your voice. Pronounce your words properly.