If a creature you control with lifelink wears an Armadillo Cloak and deals damage, first you'll gain that much life for lifelink, and then Armadillo Cloak's triggered ability is put onto the stack. This also means that multiple instances of Armadillo Cloak's triggered ability do trigger separately.

Does Lifelink use the stack?

Before the rules changes in Magic 2010, lifelink was a triggered ability, and it used the stack along with combat damage. This meant that its life gain if caused by combat, would arrive too late to prevent the player from losing the game due to combat damage from the same combat.

Can you get double Lifelink?

There's no such thing as "double Lifelink." However, you can have "double Extort," where you can pay for the Extort ability twice.