Answering out of order, but in the way that makes most sense to me:
Can I do this with RAW?
Absolutely! Gaining prestige and reputation to the level of attracting followers--of all ilk--is a facet of D&D going all the way back to OD&D. While it's been downplayed in recent adventure-path-centric variants, it's still in the DNA of D&D. The rules for Followers and Loyalty have been shoved into the DMG rather than taking their prominent places in earlier PHBs but they're there for the using, should your GM so choose. Don't expect to push this on your GM, though--it's got to be collaborative.
What level should I be before I can expect to reliably recruit and command followers?
This is highly setting/GM dependent: the question boils down to "how awesome do my powers have to be to impress NPCs?" Guidance from early versions (OD&D, AD&D, 2e) tells us that levels 8-12 were when followers would start to automatically appear, rather than being managed as hirelings/henchmen. It varied by class: Cl8, F9, R10, T10\$^1\$. All this can be swept under the rug when you remember that the original thought behind XP was that it be based off of GP, not monster-kills: you got unasked-for followers when you had the money to support them and start giving them their own missions!
Are there any useful spells to use that will help me recruit followers?
Working in the mind-set that you're out to impress yokels with your powers, you can start low with things like command and detect thoughts, super-useful for the would-be cult-leader. Except you're a warlock, so those two are off the table. Creative use of minor illusion, suggestion, and hypnotic pattern are on the table. Depending on your choices of invocations there seems to be hay to make with beguiling influence, mask of many faces, or one with shadows. And don't forget how far Raistlin got with one use of friends.\$^2\$
Really, the question seems to me not so much "can I use spells to gain followers" as it is "how much fun am I going to have gaining followers with my spells?" If your invocations have all been building on eldritch blast, then just keep preaching that this is you blazing a path for Cthulhu's return. Shriek "Ph'nglui Mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn" a few times and let the chips fall where they may. In short, let your GM know what you're going for and role-play the heck out of your character! Just about any spell, used in pursuit of Cthulhu's glory, should be useful.
Are there any feats to help me recruit or inflict madness?
Inspiring Leader and War Caster jump out if you're thinking of an aggressive/combative cult. I'd ask my DM about whether I could choose Linguist and specify R'lyehian as one of the thus-learned languages. (Who cares that no-one else on the planet can speak it?) As with the spells, I don't think so much that any particular feat is best for you mechanically--it's all in how you want to play it.
Are there any useful spells to assist in or directly inflict madness into NPCs?
Madness is directly treated in the DMG on pp.258-260. Also check out the sections on Sanity, Fear, and Horror. It's not a lot, but a good start. Especially since the DMG specifically recommends that you "consider using the Sanity score if your campaign revolves around entities of an utterly alien and unspeakable nature, such as Great Cthulhu, whose powers and minions can shatter a character's mind."
In the core set your best bet seems to be symbol, but it's both a 7th level spell and not available to your class =( Contact other plane certainly fits with the character, but as written it's not going to drive anyone else insane, just you. And you've got to wait three more character levels to access it.
Alternative routes to madness, stemming from 2015-16 Adventurers League materials
Tainted Myconids are promised in the Out of the Abyss hardcover, which spit madness-inducing spores.
Spoiled meat from the Underdark can induce madness when eaten.
Tongue of Madness is an edible fungus, though it's a bit mis-named: it causes the eater to wax on truthfully for an hour. (Think Jim Carrey's character in Liar Liar.)
Horrid Sounds can drive one mad.
Drow Warning Sign is a sigil carved into rock that, when touched, can drive one mad. I read it as a sort of glyph of madness + permanence construction.
And lastly, witnessing the rise of Demogorgon will drive you bonkers.
(All of the emphasized phrases are ones you could search for in Encounters materials.)
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(oO)
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\${}^1\$ - Recall that XP for each level varied by class in those days, so there are larger-than-obvious variations implied there. Never mind prime attribute-based XP bonuses....
\${}^2\$ - Dragons of Autumn Twilight ch.17: Beneath the plaza in Xak Tsaroth Raistlin does coin tricks and sleight of hand to gain the Gully Dwarves' admiration. Then he pulls out a scroll and "chant[s] softly 'Suh tangas moipar, ast akular kalipar.' ... 'What was that all about?' Sturm asked suspiciously. 'They are now spellbound,' Raistlin replied. 'I have cast over them a spell of friendship.'" (Bupu's one of them, p. 197 of my 80's-era paperback.) I've always assumed it was Friends, recalling that in 1e that was an AoE spell.
