If a creature with double strike also has deathtouch, does deathtouch apply to both damage steps?

Double strike causes a creature to deal combat damage in two separate steps: first the first-strike damage step, then the regular combat damage step (CR 702.4b). Deathtouch is a static ability that means any amount of damage that creature deals is considered lethal to the creature it damages (CR 702.2b). Because the creature deals damage in both steps, deathtouch applies to both instances of damage.

This is particularly powerful because after the first-strike damage step, state-based actions are checked (CR 704.3). Any creature dealt damage by a deathtouch source that received enough damage to be destroyed will be removed before the regular damage step. This means a double strike + deathtouch creature can often kill a blocker in the first step, and then deal its second hit elsewhere or go unblocked.

For example: You attack with a 2/2 creature with double strike and deathtouch. Your opponent blocks with a 5/5. In the first-strike damage step, your creature deals 2 damage. Because of deathtouch, that 2 damage is lethal regardless of the 5/5's toughness. State-based actions destroy the 5/5. In the regular damage step, your creature's second 2 damage has no blocker to deal damage to, so it deals damage to the opponent instead (if trample) or is simply wasted.

Under CR 510.2, after first-strike damage is dealt, the game checks state-based actions, so the synergy between double strike and deathtouch is fully supported by the rules. The deathtouch ability (CR 702.2c) does not limit itself to one combat damage step — it applies whenever the creature deals damage.