Focus on one out of the many accents of the English-speaking world. Choose based on the area you live in or where you intend to move to. For example, British and American accents have vast differences. Stick to the standard pronunciation and ignore regional dialects, such as Cockney of East London and Texan English.
Become an avid viewer of television shows of each region. Choose programs depicting scenes from everyday life, as the tone of news anchors and show hosts is not natural. Record the programs and review parts with speeches you have difficulty in pronouncing.
Imitate English speakers' speech, even going as far as to change your voice slightly. Do not try to use your native language's intonation when asking or expressing astonishment in English. For instance, Germans and British use a different tone when saying "Was ist los?" and "What's the matter?" respectively.
Practice speaking at the pace native speakers do. Be careful not to speak too fast and risk getting confused. Clarity in your speech is why you improve your accent after all.
Record yourself when speaking and compare your speech with what you listen to on television. Our actual voice is not what we hear as we talk, hence, a recording can help you identify mistakes only others can hear.
Practice your accent by engaging in conversations with native speakers. If you don't live in an English-speaking country, try online services, such as the video chat rooms of OnlinEnglish.