Choose a singer to imitate, either in vocal style or in songwriting style.
Study that singer's lyrics if the goal is to imitate a songwriting style. How does that singer write? Are the songs love songs, with specific kinds of declarations of love? Are the songs stories about people or places that might be more characteristic of a folk song?
Play the songs that you write for your family and friends and see if they think your compositions are close to the original singer's material.
Listen to the singer's voice. How does he or she sing? If you have chosen Michael Jackson, for example, listen to his voice: it is usually a high-pitched falsetto and his phrases are very breathy. Gloria Estefan, by contrast, is very smooth and melodic and usually adds vibrato -- the trilling of the voice -- at the ends of her vocal phrases.
Practice your singer's vocal style by singing along with his or her songs. Try to imitate as closely as you can.
Record yourself singing the person's songs. Play back the recording and listen to it to see how you did. Continue to practice this regularly.