The reason why hands shouldn't be extended fully is the fact that full extension with underhand grip might cause problems with your elbows later on. Remember not to kip or swing to gain momentum while doing the exercise. Try to eliminate this as soon as possible, as it may turn into a habbit and will be hard to unlearn. Don't drop your body rapidly after your chin is over the bar because this may lead to injuries. Also, try to engage your abdominal muscles during the exercise. Exercise should be performed on a bar or gymnastic rings. Exhale as you pull your body up and inhale when you lower yourself back.
Chin ups put the major focus on your biceps muscles because of the underhand grip, and are easier than regular pull ups. In your pull up routine, try to put overhand grip pull ups and underhand grip chin ups to balance your routine. It is recommended to put your thumb under the bar while doing chin ups.