First you need to know the melody or tune of the song fairly well enough so that it is consistent. You can sing the tune the same way each time you sing it from beginning to end at least 3 times.
Next find a part either above or below the tune at the third (either major or minor) that follows the tune. It should be in similar motion but not necessarily parallel motion. You have to allow for some variation because the harmony might not sound pleasing with the minor third so substitute the major third and vice versa. This where you have to use your taste and sensitivity.
Next find the third part which is a third above or below the second part and follow suit.
I prefer to harmonize using pentatonic harmony which usually does not use the IV or V!! of the major scale. Like in the key of C use CDE GA only this gets rid of the tritone of FB which defines a strong key center. The pentatonic melody is pleasing and rounded off with out any corners.It usually fits nicely and simply into any tune.