So, to unlock the Lightforged Draenei, you will need the following three things:
1) Have purchased Battle for Azeroth on your account.
2) Have completed the "You Are Now Prepared!" achievement. You unlock this by doing the Argus campaign quest chain. This part should not take long on a level 120 character, but the mobs can be a bit difficult for fresh level 110 characters though. The entire quest chain will take an hour or two.
3) Be exalted with the Army of the Light faction. This faction has its base on the Vindicaar, and most of the World Quests on Argus will give reputation with this faction.
You can also get rep by doing follower missions that awards the item Army of the Light Insignia, or do other reputation missions and hope you get a Greater Army of the Light Insignia. If you have a few characters, then this will definitely help speed up the process.