Does first strike damage happen before deathtouch kills a creature?

Combat damage is divided into two steps when any attacking or blocking creature has first strike or double strike. The first combat damage step (CR 702.7, 510.1) resolves damage only for creatures with first strike or double strike. Then state-based actions are checked — including whether any creature has been dealt damage by a source with deathtouch (CR 704.5g) or has lethal damage (CR 704.5j). Creatures that die here are removed before the second damage step.

This means a creature with first strike can kill a creature with deathtouch before that deathtoucher ever gets to deal its damage. Equally, a first-striking creature with deathtouch will destroy most blockers before they deal any damage back, since even 1 point of deathtouch damage is lethal (CR 702.2b).

If neither creature has first strike or double strike, there is only one combat damage step, and both creatures deal damage simultaneously (CR 510.2). In that case, a creature with deathtouch and a creature without first strike deal damage at the same time, and both can die together.

Concrete example: Your 1/1 with first strike blocks your opponent's 4/4 with deathtouch. In the first damage step, your 1/1 deals 1 damage to the 4/4. Because the source has first strike but not deathtouch, 1 damage alone doesn't kill the 4/4 (it needs 4). The 4/4 hasn't dealt damage yet. State-based actions don't kill the 4/4. In the second damage step, the 4/4 deals 4 deathtouch damage to the 1/1, which dies. If instead your 1/1 also had deathtouch, the 1 damage in the first step would be lethal to the 4/4 (CR 702.2b), and it would die before dealing any damage back.

Key rules: CR 702.7 (first strike), CR 702.2b (deathtouch — any amount is lethal), CR 510.1–510.2 (combat damage steps), CR 704.5g and 704.5j (state-based actions for deathtouch and lethal damage).