Well your right about the part where Asriel took them to the surface. However, Asriel got attacked by the humans and then picked Chara`s body up again and took it back Underground, THEN turned to dust.
That`s why there was a coffin in the castle with the soul of determination on it. When Toriel left Asgore, she took Chara`s body with her to the ruins and buried Chara. That`s the flowerbed you land on at the beginning of the game. Your landing on where Chara was buried.