Doesn't take more than five hours so if your starting this now then you still have time.
The Elsweyr expansion can be bought off the Microsoft Store. You can not play it for free with an ESO Plus subscription.
Once booted into the game, accept main Elsweyr quest and teleport to the way shrine already discovered there. Once in the Northern Elswyer region I suggest going around the main road that circles the map to discover the remaining way shrines.
Once that's out of the way wait to see a dragon icon (looks something look the one from the Mortal Kombat game series logo) appear on your map and then simply travel to it. Put some damage on it and then when it dies you should get 3% progress on your achievement tracker.
During the fight the dragon will get up and fly around, usually twice per fight. Be careful standing to close to its sides and tail, it will swing them at you. Its face will breath fire forward but has attacks that scatter every where, just watch for red circles on the ground or rolling fireballs. Fights usually take five minutes.
You can join a party that hunts dragons but unless you want the bonus XP, it's not needed. Just travel to any dragon that is up and people will be there.