Play the melody. Write down the names of the notes that start and end each measure in order to find out the melody's key. Pay close attention to the ending notes that resolve the melody and the notes that are accented. The ones that appear most often may be the name of the key.
Write down the other tones that have appeared within the measures. Compare the notes that are used in the melody with the notes that are found in the major scales of the keys named by the notes that you wrote down that were prevalent in the melody. For example, if the note "C" was the one that ended the melody, see if any of the other notes that you wrote are found in the scale of C Major, which is the one that has no sharps or flats. Keep in mind that on occasion, passing notes are played in the song that are not in the named key.
Call the name of the key the number one. (It is also called the "tonic" note.) Count to the fourth note of the scale and then the fifth. They are named subdominant and dominant respectively. All three named notes can be used as major chords that repeat throughout the song. For example, in the key of C, call the C major chord "One." F major would be the fourth or subdominant. G major would be the fifth or dominant.
Strip the melody of passing tones and unaccented notes. Scrutinize each measure and try playing the tonic, subdominant or dominant chord for it. If they don't seem to fit, use minor chords of the second, third and sixth notes of the key. For example, in the key of C, a D minor, E minor or A minor might go well with a measure.
Embellish the melody with seventh, major seventh and minor seventh chords. Use seventh chords as transition tools to resolve a phrase. Use more than one chord per measure if it sounds good to you.