There is 3 main styles of grip for the deadlift, each of them have pros and cons. Let’s review each of them:
THE DOUBLE OVERHAND GRIP:
This is the most common grip for the non-strength athlete, both palms facing the lifter (pronation), a thumb wide outside the edge. The grip stance varies depends of the deadlift style, if it normal stance or sumo stance, which will require a closer grip.
This grip is great for building forearms strength but it will limit your deadlift to the strength of your forearms, and obviously you can lift much more than your forearms can handle.
THE MIXED GRIP:
This is when one hand is facing the lifter (pronation) and the other hand is facing away from the lifter (supination). This is the most common grip used in powerlifting, it allow you to lift much more weight than using the double overhand grip, and it has the advantage that it also build strength in your forearms. The downside is that you have to alternate between training which hand is in pronation and which hand is in supination, otherwise it will lead to unbalance your muscles and later on leading to injuries. Even if you alternate this grip, you can injure yourself because when you use such grip, your body twist a little bit once you are deadlifting, and can have a potential impact on your spine.
THE HOOK GRIP
This grip is a double overhand grip but with the thumb trapped under the other fingers of your hand, this is why it is called the hook grip. This is the grip used by all Olympic weightlifters, it is the strongest grip and allow them to lift very fast without the grip being broken, and you keep a perfect posture while using this grip, no unbalanced muscles, no twist on your spine, etc. But the downside is that it completely take off the work of the forearms and you don’t build any strength in your forearms with this grip. So my advice is to combine the grips, personally I use the double overhand grip when I warm up to build some forearms strength, then I use the hook grip when weights become heavy. But you also can use the mixed grip and alternate it, it depends on how you feel. Try to change of grips sometimes and check what the best suits for you.