If a creature has both first strike and deathtouch, does it kill a creature before regular damage is dealt?

Combat damage is dealt in two separate steps when any creature has first strike or double strike (CR 702.7). In the first combat damage step, only creatures with first strike or double strike assign and deal their damage. Then, if any creatures without first strike (or with double strike) remain, a second combat damage step occurs for them (CR 509.5).

Deathtouch means that any amount of damage a creature deals is considered lethal to the creature that receives it (CR 702.2b). State-based actions check for lethal damage and destroy the damaged creature (CR 704.5g). These state-based actions are checked at the end of the first combat damage step, before the second step begins.

So if your first-strike + deathtouch creature deals even 1 damage to a blocker in step one, that blocker is destroyed by state-based actions before it ever gets to deal damage back in the regular (second) damage step. This makes the combination especially powerful, as the attacker can kill almost anything without receiving damage in return.

Example: Your 1/1 with first strike and deathtouch blocks a 5/5 vanilla creature. In the first strike damage step, your 1/1 deals 1 damage to the 5/5. State-based actions destroy the 5/5 (deathtouch makes 1 damage lethal). The second damage step occurs but the 5/5 is gone, so your 1/1 takes no damage at all.

Note: if the blocker also has first strike, both creatures deal damage simultaneously in the first step, so the deathtouch creature does not get to dodge damage in that scenario.