Does double strike cause a creature to deal damage in both the first strike and regular combat damage steps?

Double strike (CR 702.4) causes a creature to deal combat damage twice: once during the first strike combat damage step and again during the regular combat damage step. This is explicitly defined in CR 702.4b, which states that a creature with double strike assigns damage in both steps.

The combat damage phase works as follows (CR 510–511): if any attacking or blocking creature has first strike or double strike, a first strike combat damage step occurs before the normal combat damage step. Creatures with only first strike deal damage in the first step and skip the second. Creatures with double strike deal damage in both steps. Creatures with neither ability only deal damage in the regular step.

This means a double strike creature effectively gets two separate opportunities to deal damage, which can kill blockers in the first step and then still deal damage (to the player or planeswalker, or another blocker) in the second step.

Example: You attack with a 2/2 creature with double strike. Your opponent blocks with a 2/2 creature. In the first strike step, your creature deals 2 damage, destroying the blocker. In the regular combat damage step, your creature deals 2 damage again — but since the blocker is gone, that 2 damage goes to your opponent's life total instead.

This is one of the most powerful aspects of double strike: it can effectively "punch through" a blocker and still deal damage to the defending player in the same turn, something a creature with only first strike cannot do.