The grip I use is one in which the putter is placed in the hollow part of the left hand (for right-handed golfers). My right hand is underneath enough so that my right thumb rests on the left side of the handle, as illustrated above.
BALL POSITION AND STANCE
Make sure your ball position is toward the forward heel and your toes are parallel to the target line. Your eyes should be over the ball. This helps not only in being able to aim properly, but also with seeing the line of the putt more easily.
INSIDE DOWN THE LINE
I have had the best results and most consistent ball roll using a putter path that goes slightly inside on the backstroke and then down-the-line on the forward motion. Make sure when making the backstroke that the putter blade stays square and does not fan open. There is a drill I recommend to help achieve this "Inside, Down the Line" path. Hold the putter in the right hand only (as illustrated) and hit putts to acquire the feel for this system.
After hitting some putts with the right hand only (for right-handed golfers), go back to putting with both hands on the handle but make strokes with a hand pressure of 80-percent in the right hand and 20-percent in the left hand. In other words, putt predominantly with your right hand. As noted at the beginning, putting is a very personal part of the game, and different teachers teach it in different ways. But this is the putting approach that has worked for me and my students.