There are 11 exotic weapons and 4 armors per class added in the Forsaken DLC. You will have to have some luck to complete this achievement, but some are available through quests. I will start with the ones through RNG first.
Cerberus +1, Trinity Ghoul, Two-Tailed Fox, Black Talon, and Queenbreaker.
One-Eyed Mask, Ursa Furiosa, Heart of Inmost Light, and Antaeus Wards
Shards of Galanor, Oathkeeper, Sixth Coyote, and Gwisin Vest
Contraverse Hold, Chromatic Fire, Phoenix Protocol, and Geomag Stabilzers
There are also four available through quests, and two that only drop from specific things in the Forsaken DLC
Wish-Ender- Available through a quest that starts in the
Ace of Spades-Available through quest you receive at the end of the Forsaken campaign avenging
The Chaperone-A quest that begins at the end of the campaign given by Amanda Holiday
Malfeasance-From a quest that begins by killing a special boss in the new game mode, Gambit. This boss spawns very rarely but will have a higher chance at spawning on the 3rd and final week of the corruption cycle.
Lord of Wolves-Has a high chance of being dropped from the Wanted bounties from Spider on the Tangled Shore.
A Thousand Voices-Only available through the chest at the end of the Last Wish raid after killing
in the chest room. Only a chance of having it drop.
You only need 10 total, and they carry between characters. I got mine with 6 weapons, 3 Titan Armors, and 1 of the Hunter Armors. For the ones that drop by RNG I would recommend doing any bounties, or daily and weekly objectives that drop Powerful Engrams. If you have any questions feel free to leave a comment on this guide.