You should start by learning the phonology of the English language and its main dialects (namely Standard British English and General American). You need to know the theory because if you try to rely on your perception it will fool you. You need to know what you have to pay attention to. For example, if your L1 is a Romance language you have to be aware that vowel length is important in English, and that diphthongs are a single phoneme. Things like these you'll never be able to figure out yourself, they need to be taught. Once you know the theory in and out, you have to get a lot of exposure to the accent that you're interested in. Lastly, you start to imitate what you hear. Bear in mind that it takes a lot of practice to get a foreign accent right.
