What is this dry skin on my Achilles tendon and heel?

Possible fungus. Dry cracked heels could be a sign of a fungal infection. Other causes include constantly wearing open heeled shoes. You can try otc Lamisil (terbinafine) cream twice a day. A skin culture may be helpful if it doesn't improve with otc meds.
I don't know. It could be an irritant or allergic reaction due to clothing or shoes or exercise; or it could be an unusual presentation of athlete's foot; or it could simply be dry skin. Someone (preferably a dermatologist) will need to physically look at it to give you a more precise answer.