To activate most dual attacks, the Assassin must first land a successful off-hand attack. Dual attacks always result in a double strike and the dual attack's specific effects will trigger on each hit.
For example, Twisting Fangs, at 16 Critical Strikes, will show up as +21 damage if you hover your mouse over the skill in-game. But since you'll be hitting twice, you'll actually do a total of +42 damage if both strikes hit.
It is important to note that dual attacks do not necessarily always hit twice. They have independent miss chances, are blocked independently, and there is about 1/8th of a second time period between the first and second attack, which means that it can actually be half way interrupted. For example, if you used Twisting Fangs against someone protected by Shield of Judgment, it would only do +21 damage because you would be knocked down after the first hit, preventing the second hit from occurring.
Dual attack effects (such as conditions) go off on both the first hit and the second hit, but don't stack. So if you were interrupted half way through Twisting Fangs, you'd lose that extra +21 damage attack, but the target would still be afflicted with both Bleeding and Deep Wound because of the first hit. Or, in cases where the enemy had Riposte up, the first hit would be blocked, but the second would still put Bleeding and Deep Wound on them.
If a dual attack is used out of sequence, it will consume energy and miss, but will recharge instantly.
While dual attacks are usually the final link in an Assassin's attack chain, there are still skills that can lengthen this chain. Impale and Assault Enchantments are both skills that follow a dual attack and Moebius Strike is an off-hand attack requiring a dual attack, thus enabling the attack chain to be looped back around to a Dual Attack again.