Destiny 2

This is available after completing the main forsaken campaign. Talk to Petra and the Spider and you will be given a pursuit involving the Broken Awoken Talisman. The first step is to complete 3 lost sectors (per koolkong94: There are 5 lost sectors in Tangled Shore. 2 in Thieves' Landing, and one in every other major zone. You need the ones in Jetsam of Saturn and Four-Horn Gulch and the southern one in Thieves' Landing) the one you don't need is the one that requires you to take the elevator to the top. Once you complete the 3 lost sectors you next have to kill taken enemies.

You can do this however you want, including the Lake of Shadows strike. I personally did it by going to Io and running the lost sector right across from the fast travel in the Rupture. Takes about 6 runs through the lost sector or 2 runs through the strike. Now the (potentially) hard part.

The next step involves completing an ether harvest public event which only spawns in the four-horn gulch area. The only problem is there are 3 different public events which can spawn in this area. I had to wait through 3 public events before it finally spawned. The event involves killing 3 chieftains and then killing the abomination. I actually walked away while waiting and came back to find that 2 chieftains had already been killed before I joined in, luckily the quest step was still completed at the completion of the public event.

Last but not least you will be given an actual story mission, and once completed you have full access to the Dreaming City. I believe the achievement actually pops when you first arrive in the city. Have fun, the Dreaming City is tough!

Up to date solution as of June 2020

You no longer need to play any of the Forsaken campaign to get this achievement. Just visit the Dreaming City (it's a tower off to the right on the Director screen) and it will pop.

EDIT: More recent updates have made the Dreaming City accessible without owning Forsaken—but you still need to own Forsaken to pop this achievement.