This is how my priest explained it to me during my catechism (if I recall correctly):
Christ died, this means that His Body and His Spirit were separated. The Godhead remained united to both His Body and His Spirit at all times. He was God when He descended to Hades. His Body in the Tomb did not decay.
Christ did everything without separation between His Humanity and His Divinity. We can't say Christ the God did this, and Christ the man that. There was no separation between His humanity and His divinity. However, when He hungered it was because of His humanity, not His divinity. This doesn't mean He hungered separately from His divinity, but His hunger was an attribute coming from His humanity rather than His divinity. Likewise He died in that His Body and Spirit were separated, not that He died in His divinity... But just like His divinity did not separated from His humanity when He hungered, rather He, who is human and divine hungered, even though the hunger was because of His humanity and not His divinity, likewise He remained fully Human and God when He departed, with both His Body and His Spirit being united to the Godhead at all times.