There are many conflicting reports on internet regarding how to kill and spawn Taken Champions on Earth in Destiny: The Taken King. This guide provides step by step procedure on Spawning and Killing Taken Champions on Earth in Destiny: The Taken King.
Launch patrol on cosmodrone, head out to skywatch and make sure you are not the only one there. Eventually you will see “the taken are corrupting the area” on the bottom of your screen. This means Taken will now start spawning around you (not always the same spot keep in mind) You must kill these taken mobs in quick fashion and this will proceed to spawn taken lieutenants, kill the liuetenants as quickly as possible and if done correctly a taken champion will spawn in Skywatch.
Note: Not to try this out solo until and unless you have a good gear at level 40 and at 220 light or so.