If a creature has both lifelink and deathtouch, does it gain life when killing a creature with one damage?

When a creature has both lifelink and deathtouch, both abilities apply to every point of damage it deals. Lifelink causes damage dealt to also cause its controller to gain that much life (CR 702.15b), and deathtouch means any amount of damage it deals is sufficient to destroy the creature it damages (CR 702.2b).

So if your 1/1 with both lifelink and deathtouch deals exactly 1 damage to a 10/10 creature, two things happen at once: you gain 1 life from lifelink, and the 10/10 is marked with lethal damage from deathtouch. State-based actions then destroy the 10/10 when they are next checked (CR 704.5g).

It does not matter how much toughness the defending creature has — deathtouch makes even 1 damage lethal. This also matters for combat damage assignment; a creature with deathtouch only needs to assign 1 damage to each blocking or blocked creature to be considered lethal for the purposes of assigning damage to additional creatures (CR 510.1d).

Example: You attack with a 1/1 Snake that has both deathtouch and lifelink. Your opponent blocks with a 5/5. Your Snake assigns its 1 damage to the 5/5. You gain 1 life (lifelink), and the 5/5 is destroyed at the end of combat damage (deathtouch). Your Snake dies to the 5 damage back, but the trade and the life gain both happen.

There is no interaction problem between the two abilities — they apply independently to the same damage event, so both always trigger or apply whenever the creature deals damage.