Rest the end portion of the atlatl between your thumb and index finger with the nock facing upwards. You should hold the atlatl tight enough that it stays in your hand during the throw but you can still move your wrist freely.
How to Hold a Dart
Put the tip of the dart on the ground and rest the end on your shoulder. Nock the dart and then grip it like a pencil. Lift the tip of the dart off the ground, if the end does not come off the nock then you are gripping it tightly enough.
How to Position Your Body and Throw
With both feet pointing towards the target, place the foot on your dominate side back and at a 45 deg angle with the other foot. Raise the atlatl and dart to eye level. Keeping your wrist straight, bring your arm back as comfortably as possible while keeping the dart lined up with the target. Lean back on your dominate foot then step forward putting the weight on your opposite foot while you slide your arm forward. When your arm is almost completely extended forward, snap your wrist downward -- the atlatl should end up pointing towards the target.