Go get your first 8 Requiem mods. Steel yourself: this is the most painful part of the whole process. Once you have the first 8, replacing the ones you use isn't that bad.
If you hate the whole kuva extractor thing, you can do the kuva flood missions without doing the kuva extractor and you'll still get the relics. In public games people will usually want to do the extractors and they don't take very long (and the other people might be there for the kuva, not the relics!), so don't be a jerk! But you can solo the missions quickly or, if the group can't find the extractor, let them know that they can just go to the end instead of searching all over for it if they only want the relic.
Waiting for the Rescue fissure is probably the fastest way to open a bunch of relics.
Spawn your lich by doing a grineer mission over 20. Spy missions seem buggy (at least they were a few days ago). Exterminate is good. Adaro is popular. Your screen will flash like Stalker is attacking, you'll get a transmission, and then you'll get a marked enemy to go kill.
To kill a lich, you need to have the right three Requiem mods in your parazon in the right order.
The lich controls the parts of your map with the red blobs on them. If you do the lich missions on them, you'll run into the liches themselves and Thralls.
Thralls are marked with a red icon. They'll spawn periodically throughout these missions. Kill them, then use the Mercy finisher on them to get Murmurs. Once you get enough Murmurs to fill up the meter, when you extract it will reveal one of the three Requiem mods (although it doesn't tell you which slot it goes into). Every Thrall you kill increases the Anger of your lich, making it more likely to spawn into a mission (this carries over between missions).
When the lich appears, you'll see it has three health sections. After you damage it enough to empty the first health section, the lich will kneel down and begin regenerating health. If it refills the bar, it will get back up. While it's down, you can use a Mercy finisher on them - if you have the right Requiem mod in the first slot of your parazon, you will destroy the first health bar and repeat the process with the next (after the third, you will defeat the lich); if you don't have the right Requiem mod, the lich will kill you and leave.
You should always try to kill your lich every time it appears!
The two slowest ways to kill your lich are:
Trial and error without Murmurs.
This will take up to 19 tries, and without killing Thralls (which would give you Murmurs) the lich will spawn very rarely.
2. Ignoring the lich while farming the Murmurs from Thralls, then using trial and error to guess the correct order of the mods.
This will take a lot of time to kill all the Thralls, especially now that the third reveal takes much longer than the first two. Once you've revealed all three of the mods, it will take you up to 4 tries to figure out the correct order of the mods.
Almost no one seems to be trying 1, but a lot of people seem to be trying 2! Don't do this! Yes, the lich will probably kill you, but it's worth it!
You should use the lich to narrow down the possibilities while you farm murmurs. Also, if you're playing with other people and ignoring your lich, you're screwing the other players over because only one lich can be spawned in at a time. The recent change makes it even more disadvantageous: the third murmur reveal takes longer than the first two (so you have more time and it's even better if you can guess it before getting to the third reveal) and you're not getting the extra murmurs from fighting the lich.
Instead, you want to be gathering murmurs and also gathering information every time the lich spawns. The way to do this is:
Start with 3 random Requiem mods in your parazon. You don't really have a choice - you don't know anything yet.
As you farm thralls, the lich will invade. Fight the lich every time it invades. After you fight the lich, change the Requiem mods in your parazon.
If you got lucky and the first mod was correct, now you know the first mod (don't change it!), and you also get to see if the second mod was right too.
If you were wrong, change out the first mod: you know it isn't that one.
This should be straightforward - it's just what you would do if you were trying to figure out the combination through trial and error.
4. When you learn one of the three mods from the murmurs, there are two possibilities:
If you've already tried the revealed mod in the slot you're working on, put it in the next slot, and continue trying new mods in the slot you're working on.
If you haven't tried the revealed mod in the slot you're working on yet, put it in that slot. Next time you run into the lich, you'll either find out it's correct and start working on the next slot, or you'll find out it was wrong, in which case you should move it up to the next slot.
If you do this, you will often be able to kill the lich before the final Requiem mod is revealed from Murmurs, and even if you haven't figured it out and do get to the third Murmur reveal, you will always know something about the order of the mods.