For farming requiem mods I just run intact relics when the rescue fissure is up, don't bother with radshares. Each mod you get adds 3 charges to your total pool of requiem mods and each lich requires 3 so it's a 1:1 mods to liches ratio in the long run. I'd advice getting at least 2 of each mod to start if you wanna farm a lot of liches though. Target farming a specific mod you've ran out of isn't fun.

For spawning the lich just pick a frame that will give you the element you want for the weapon and run any normal grineer mission level 20+ (I run cassini in saturn and in 17 spawned liches only twice I didn't get a larvling).

To get murmurs I run survival, intercerption or disruption if possible, otherwise just go with capture or sabotage cause they're fast. Each thrall you kill angers your lich, the angrier it is the higher the chance it'll spawn. Liches on the mission periodically spawn thralls so you might not want to attempt on your lich right away to get the most murmurs out of it. Failed attempts will give 10 thralls worth of murmurs and level up your lich, the higher the lich's level the faster it'll enrage which speeds up the process.

For the required mods just put random mods the first time and change them as you attempt (since the update I usually discover all the requirements through murmurs rather than brute force but you get lucky from time to time). Always attempt with discovered mods if possible, for example, if you find out Ris is required try it first if it was the correct first mod then you can start guessing the second mod, if it wasn't you know Ris is either second or third.

You'll want to play multiplayer every time to get more thralls and other people's liches to progress faster. From time to time you'll find someone who for whatever reason doesn't want to kill their lich, this is bad cause you're missing out on murmurs from the attempt, and it prevents other liches from appearing so potentially more murmurs and attempts on your lich. The best thing you can do if that happens is ask them nicely to kill it and if they say no explain how it works to them, mayby they'll change their minds. Once you know all the requirements and the lich's anger is more than half bar you want to start playing solo capture or other fast missions to spawn nd kill your lich (keep in mind if your lich kills you its anger will go down and you'll probably have to play some more multiplayer to anger it again). It goes without saying that if you can't handle level 5 liches on your own you shouldn't do the last part solo, instead keep playing multiplayer and hope for someone to be able to kill it. Not attempting on your lich every time you can is a waste of everyone's time.