Does double strike cause lifelink to trigger twice?

Lifelink is a static ability (CR 702.15a), not a triggered ability, so it does not 'trigger' at all. Instead, it modifies damage events: whenever a source with lifelink deals damage, its controller gains that much life as a result of that damage being dealt (CR 702.15b).

Double strike (CR 702.4) causes a creature to deal combat damage twice — once in the first combat damage step and once in the second combat damage step (CR 702.4b). Because lifelink applies to each instance of damage dealt, a creature with both lifelink and double strike will cause its controller to gain life twice — once for each combat damage step — but this is not because lifelink 'triggered twice.' It is because damage was dealt twice, and lifelink applies to each event.

The total life gained equals the total damage dealt across both steps. If the creature's power is 3, you deal 3 damage in the first step (gain 3 life) and 3 damage in the second step (gain 3 life), for 6 life gained total.

Example: You attack with a 3/3 creature with double strike and lifelink. In the first combat damage step it deals 3 damage — you gain 3 life. In the second combat damage step it deals another 3 damage — you gain 3 more life. You gain 6 life total, not because lifelink triggered twice, but because damage was dealt in two separate steps.

Key rules: CR 702.15 (lifelink), CR 702.4 (double strike), CR 702.4b (two combat damage steps), CR 510.1 (first-strike/double-strike combat damage step).