Thing in the Ice enters the battlefield with four ice counters on it.
Whenever you cast an instant or sorcery spell, remove an ice counter from Thing in the Ice. Then if it has no ice counters on it, transform it.
For more information on double-faced cards, see the Shadows over Innistrad mechanics article (http://magic.wizards.com/en/articles/archive/feature/shadows-over-innistrad-mechanics).
When Thing in the Ice’s triggered ability removes the last counter from it, Awoken Horror’s ability will trigger and resolve before the spell that caused Thing in the Ice’s last ability to trigger.
Removing all ice counters from Thing in the Ice some other way will not cause it to transform. You’ll need to cast an instant or sorcery spell and cause its last ability to trigger.