Can a creature with first strike and deathtouch kill a creature before it can deal damage back?

Combat damage is dealt in steps. Creatures with first strike (or double strike) deal their damage in a special first combat damage step that occurs before the normal combat damage step (CR 702.7b, 510.1). Creatures without first strike deal damage only in the normal step.

Deathtouch means that any amount of damage that creature deals to another creature is considered lethal (CR 702.2b). State-based actions are checked after each combat damage step (CR 704.3), so a creature dealt even 1 point of deathtouch damage is destroyed before the second combat damage step begins.

This means a creature with both first strike and deathtouch will deal its damage in step one — and because that damage is lethal thanks to deathtouch — the defending creature is destroyed by state-based actions before it ever gets to deal damage back in the normal combat damage step.

Example: Your 1/1 Viridian Longbow Archer (equipped with deathtouch) attacks. Your opponent blocks with a 5/5 Dragon. In the first combat damage step, your 1/1 deals 1 deathtouch damage to the Dragon. State-based actions destroy the Dragon. The normal combat damage step occurs, but the Dragon is already gone and deals no damage to your creature.

Note: if the blocking creature also has first strike, both creatures deal damage simultaneously in the first step, so the deathtouch creature does not get a timing advantage over that specific blocker.