Does first strike damage happen before deathtouch is checked for state-based actions?

First strike (and double strike) creatures deal their combat damage in a special first combat damage step that occurs before the regular combat damage step (CR 702.7b, 510.1). State-based actions — including checking whether a creature has been dealt lethal damage — are checked after each combat damage step, not during it (CR 704.3).

So yes, a creature with first strike can kill a creature with deathtouch before that deathtouch creature ever gets to deal damage in the regular damage step, provided the first striker's damage is lethal (or the deathtouch creature doesn't also have first strike or double strike).

Critically, deathtouch means that ANY amount of damage dealt by that source is considered lethal (CR 702.2b, 704.5g). So if the deathtouch creature does deal first strike damage (because it also has first strike or double strike), even 1 point of that damage will destroy the creature it damaged when state-based actions are checked.

Concrete example: A 2/2 with first strike blocks a 1/4 with deathtouch. In the first combat damage step, the 2/2 deals 2 damage to the 1/4. State-based actions are checked: the 1/4 has 2 damage but has 4 toughness and is not destroyed. In the regular damage step, the 1/4 deals 1 deathtouch damage to the 2/2. State-based actions are checked: because of deathtouch, any damage from the 1/4 is lethal, so the 2/2 is destroyed. The 1/4 survives.

If instead the deathtouch creature also had first strike, both creatures would deal damage simultaneously in the first combat damage step — the 1/4 deathtouch/first strike deals 1 damage to the 2/2 (lethal via deathtouch), and the 2/2 deals 2 damage to the 1/4 (not lethal). State-based actions then destroy the 2/2, and the 1/4 survives.