The following is how some people describe how to hold your crochet hook, and also your left-hand "tension":
Always keep thumb and forefinger of your left hand near the stitch on which you are working.
Holding The Crochet Hook
There are two methods of holding a crochet hook. Try them both to see which
one feels most comfortable to you.
Place the hook in your right hand, holding it between your index finger and thumb, as you would hold a pencil.
Hold the hook in an "overhand" clutch, similar to the way you might hold a tennis racquet, or a spoon while making cake batter.
(I apologize for not noting this link down when I found it.)
To crochet easily and successfully, you need to hold the yarn and the hook comfortably, with enough tension on the yarn, so that when you draw the hook around the yarn, it stays firmly in the lip of the hook. Most people choose to wrap the yarn around their fingers, and some make an additional wrap around their little finger – choose whichever yarnholding method works best for you.
Position of the hands
Hold the work in the left hand, between the thumb and forefinger, yarn is taken over the forefinger and middle finger, under the third finger and around the little finger. The crochet hook is held in the right hand (as you would hold a pencil) between the thumb and forefinger and pressing on the middle finger.