Listen to the music two to three times from beginning to end to become familiar with it. After this, you can go back and focus on the complex parts.
Sing the melody out loud to continue familiarizing yourself with the music. If necessary, go back and listen to the recording until you're able to sing it from beginning to end.
Jot down basic impressions about the piece in general. This makes you ask yourself what kind of emotions it draws out (e.g. happiness, sadness, relaxation), which makes the re-composing process easier.
Write notes about actual structure of the song. For example, determine where the song repeats, pauses, changes key or tempo. This helps make the re-composing process easier since it requires you to examine the piece.
Fill in the key and time signature you've chosen on a blank piece of sheet music. This is chosen based on the above research and your vocal or instrumental range.
Write down the melody, note by note. Play it back on an instrument or by singing it and make any necessary adjustments.
Add the base line. If it's unclear, go back and listen to the music and write down what you hear, note by note. According to SeventhString.com, the base part is important because it allows you to flesh out the chord with more ease.
Look at the melody line and the base line. Based on those two notes, you should be able to determine each individual chord. Once you know the chord, write down the tenor and/or alto line.
Finish the piece by adding all the included pauses and time or key signature changes. Listen to the piece once you're finished and make adjustments, if necessary.