All damage assignment has to be done prior to applying any double damage effects.
If a 4/4 blocks the Nova Chaser, you must assign 4 out of the 10 damage to the creature to assign lethal. Then the rest of the 6 is assigned to the defending player.
When combat damage is dealt, the damage is then doubled. So the 4/4 is dealt 8 damage, Nova Chaser is dealt 8 damage, and the defending player is dealt 12 damage.
Note that even though Nova Chaser has trample, the additional 4 damage dealt to the 4/4 doesn't trample through. You cannot reassign that damage to the defending player.