Unlike a full swing where there are prescribed ways to hit a good shot, putting is another story. Aside from keeping your body still, your wrists unbent and swinging the club like a pendulum, a golfer should adopt the attitude of "If it feels good, do it." For that reason, there are myriad grips that golfers use to get the ball in the hole.
Adopt the "reverse overlap grip" and you will putt like the vast majority of golfers. First, grip the putter with your left hand. (Right hand if you are left-handed.) Your thumb should extend down the shaft, and the "V" created by your thumb and forefinger should be pointing over your right shoulder. Then grab the club with your right hand so the thumb fits neatly in the palm of your right hand. Take the forefinger of your left hand and put it on top of the pinkie of your right hand. The purpose of this grip is to make your hands move as a unit.
Use the same grip with your putter as you do with other clubs. Most people use the so-called "Vardon grip" that is similar to the grip described above except the pinkie of your right hand is placed over the forefinger of your left. Many golfers feel comfortable using this grip although the hands do not work in unison as much as they do with the "reverse overlap grip."
Grip the putter as you would a baseball bat with all five fingers of both hands placed on the club. While many players make sure their hands are abutted, some separate their hands to give them more feel of the putt.
Base your grip on the length of your putter. Some players, including professionals, use a longer putter with its butt firmly placed on their lower chest because they feel it creates more of a fulcrum for putts. They then will use one of the grips described above. Or they will adopt the so-called "claw grip" where they will place their left hand on the putter. But they will hold the putter with only the thumb and forefinger of their right hand, with the back of their right hand facing outward. This is a relatively new grip, and those using it feel it keeps their wrists firmer during putts.