When a permanent is being made to copy another, does it count as entering the battlefield?
No, it will not trigger the Goliath since nothing is entering the battlefield, a creature is just changing characteristics.

What you can do, however, is wait for a new creature to enter the battlefield and trigger the Goliath. With the trigger on the stack, you can then Polymorphous Rush the triggering creature, and the Goliath will get counters based on the new power. This is because the Goliath's ability doesn't check to see what the power of the creature is when it enters, it checks as the ability is resolving.