Acquire recordings and videos of people native to the region with the desired dialect. There are many options for acquiring these records, such as online in the form of YouTube videos or included on CD with an acting dialect book.
Listen and watch for the positioning of the vocal apparatus. How you use your mouth, lips, jaw, tongue and soft pallet play a very key part in how dialects are formed. The standard British accent, or BBC British, is spoken with a fairly tight jaw, corners of the mouth tensed and the mouth in a closed position. The London Cockney is spoken with a loose jaw, open mouth, relaxed lips and relaxed corners of the mouth. Changing the way you use your vocal apparatus goes a long way to creating a believable dialect.
Listen to the sounds of the words being spoken and try to mimic them. Each dialect uses slightly different methods to pronounce certain vowels or consonants. An upper class English accent tends to drop the final R sound and any Rs before another consonant. Listen and mimic the changes in phonetics to help duplicate the accent.
Listen for stress patterns in the desired accent. The stress pattern is the rhythm in which the dialect is spoken. The English language has a fairly random stress pattern and there are particulars for virtually every dialect. For example listen to how different English dialects pronounce the word garage. In standard British you'll often hear it stressed as gar'age.
Listen for the pitch patterns. The pitch pattern is the music or the intonation of the dialect. Stressed syllables in English are spoken with a higher pitch then unstressed syllables. You'll hear the greatest difference in pitch patterns when comparing standard British to elsewhere in the United Kingdom, such as Welsh or Scottish. Developing a proper pitch pattern will help your dialect have the proper feel to it.