If you happen to already have captains and/or warchiefs branded, this will be much easier to accomplish, but this strategy is for when you need to start from scratch.
I started with the murders portion. Start a Test of Wisdom and you should be able to find the captain Lorm pretty close by. I abused him in several other challenges, so we go way back. He's afraid of fire, so shoot an arrow at the campfire he's standing near to burn him and you can quickly brand, dominate, and command him. All you have to do is have him *start* a murder at the Sauron's Army screen; you don't even have to have him complete it. Then not too far from him, you should be able to find the captain Lugdash. Blow up the campfire he's standing around and it should kill all his compadres. A few arrows to the head should bring him down to green and then you can do the same thing you did with Lorm. Quit out and repeat 5 times and Bob’s your uncle. (Note, sometimes the above two captains are not there and sometimes the campfire isn’t there either, but this should work most of the time).
Now for the riots. With Lorm and Lugdash in your command, choose a warchief to go after, as only warchiefs can start riots. I picked the fifth warchief, so the one all the way on the right of the Sauron’s Army screen. The idea is you want to turn one or both of your branded captains into the warchief’s bodyguards and then kill the warchief so that one of them can take his place as the new warchief. You’ll have to go through an easy initiation quest to have each of them become a bodyguard, and once that’s done you can command your captain(s) to betray the warchief. If you’ve picked a warchief with an easy weakness, this shouldn’t be too difficult, but be aware you’ll have to help your captain out by killing all the warchief’s friends and possibly the warchief himself. Once the warchief is dead, your branded captain will be promoted to warchief. Now you can dominate your new branded warchief and command him to start a riot against a random captain. You'll get credit for that one, so now have one of your branded bodyguards murder that same captain, which cancels the riot and you can rinse and repeat. After you've started 5 of them, the achievement should pop. (Edited that last part thanks to BTrammell21).
If you see any glaring holes or ways to improve this, please let me know and I’ll be happy to edit with credit. I just want to help people get this challenge done in as little time as possible. As far as I can tell, it’s the last challenge they’re releasing after 19 grueling weeks. No more, please.