Or can you place that Enchantment Creature as a Creature on the battlefield after the Creature it was attached to dies?
Depends on the situation. Normally, when a creature dies (goes to the graveyard) or gets exiled, all attached enchantments are moved to the graveyard as well.
704.5n If an Aura is attached to an illegal object or player, or is not attached to an object or player, that Aura is put into its owner’s graveyard.
Unless you have a bestow enchantment:
702.102e If an Aura with bestow is attached to an illegal object or player, it becomes unattached. This is an exception to rule 704.5n.
Of course, that has also has its exceptions, for example when a player casts End Hostilities, then all creatures, and its attached permanents are destroyed.