For this, there is no actual quest called 'Echoes of Oryx'. Rather, you have to complete a series of other quests. There are a total of six quests to be completed.
You get the quest Taken War: Earth from Cayde 6 after reaching level 40 and completing the mission 'The Legion's Beachhead'. The first step is completing the mission 'Blighted Coven'. Return to Cayde 6 and he will task you with defeating 3 specific Taken champions on Earth.
So, you can't just farm the same one three times. The Divide and the Rocketyard are right next to each other. Skywatch, on the other hand, is over by itself. From where you start when going on Patrol, Skywatch is two areas to the left, just after the Mothyards. The Divide is the first area to the right, and the Rocketyard is the area north of it.
If you don't know how to spawn Taken champions, you basically have to sit around until you get the notification 'Taken forces are corrupting the area'. After a while, it will tell you that a Taken Lieutenant has spawned. A total of three will end up spawning, and you have to kill all three of them before the event ends. If you see the words 'the taken corruption subsides' then you've failed, although sometimes if you have on Taken Lieutenant left and kill it right as it says it, then it may still spawn the champion.
Of course, it is random what public event you get, and whilst taken incursions are the most common, you may get others either instead of or at the same time as the Taken event. For instance, I was struggling against the champion in The Divide, and then the Fallen Walker event started and the Taken and Fallen starting fighting each other and made it so I could take the champion and walker out quite easily. Also, it is worth mentioning that if any public event other than a taken event occurs in Skywatch, then it will spawn Urzok, the Hated; a Hive Ultra with a ridiculous shield.
Anyway, once you've killed those three champions, head back to Cayde 6, and he will give you the mission 'Kings of Decay'. Go back to Cayde 6 afterwards to complete this quest. For 'The Taken War: Venus', you need to first complete the mission 'A Cry For Help'. Speak to Ikora Rey to get this quest. This is probably the easiest of the Taken Wat quests, since you don't have to faff around killing Taken champions. The first step requires you to complete the mission 'Paradox'. Go back to Ikora Rey, and she will tell you to gather 30 Chronoshards, which will drop from Taken Vex units. Go back to Ikora Rey, and she will task you with completing the mission 'Entropy's Pinnacle'. Go back to Ikora Rey to complete this quest.
After completing the first or second of the Taken War quests, Eris Morn will you give the quest 'Blighted Worlds', which basically requires you to complete the Taken War quests apart from Petra's. 'The Taken War: Mars' quest can be obtained from Commander Zavala after completing 'The Taken War: Earth' and 'The Taken War: Venus'. The first step only requires you to kill 6 Taken Lieutenants, which you can feasibly get in just one Taken event. Instead of trying to kill the champion, just focus on the Lieutenants that spawn in with it.
Once done, head back to Zavala and he will give you the mission 'Fear's Embrace' back on Phobos. This can be quite the pain in the arse as the final encounter against the Echo of Oryx can be rather intense, since blight spots will appear, spawning in enemies, and poison gas gets released on the lower level after a certain amount of time. The final Taken War quest, 'The Taken War: Petra' can be obtained from Petra after completing 'The Coming War'. You basically are tasked with collecting items that drop from Taken enemies, and killing Taken champions. First on Earth, then Venus, and lastly on Mars.
The last quest you have to do for this achievement is called 'The Road to King's Fall', which is given to you by Eris Morn after completing 'Blighted Worlds' (and, by extension, the three Taken War quests from Cayde 6, Ikora Rey, and Commander Zavala). The first step is completing the mission 'Summoner's Circle' on the Dreadnaught. Go back to Eris Morn, and she will give you the second step, which basically requires you to complete the whole King's Fall Raid, as you have to kill The Warpriest, Golgoroth, and Oryx. Once done, head back to Eris Morn to complete this quest and get the achievement.