I don't personally have an alter (yet), but I would recommend to look on Reddit at r/lovecraft (https://reddit.com/r/lovecraft) for some ideas for Cthulhu alters. Or you can just look for some Occult alters and modify them to be Cthulhu-ish.
I have been curious about this too I want to make a great Cthulhu alter space.
I reckon it can be thought of this way; people generally follow the same traditions for holiday festivities, like leaving cookies and milk for Santa Claus, but each household tends to do it in a slightly personalized way that makes it different, such as the hiding of the pickle, moving of an elf, using an angel instead of a star, etc. With this parable I mean to say that there is no ONE way to create an altar. To do so, much like the creation of idols or masks, is a very personal task and it is up to the cultist to sort of create whatever they are comfortable with and feels adequately serves them in their ritual needs. Here, the general idea to follow would be the Archetypes, but how you create a space reserved for them is entirely up to you and your discretion. Have fun with it, and express yourself as a cultist!
Well, one way would be to do it the way HPL mentioned in "The Call of Cthulhu":
"In a natural glade of the swamp stood a grassy island of perhaps an acre’s extent, clear of trees and tolerably dry. On this now leaped and twisted a more indescribable horde of human abnormality than any but a Sime or an Angarola could paint. Void of clothing, this hybrid spawn were braying, bellowing, and writhing about a monstrous ring-shaped bonfire; in the centre of which, revealed by occasional rifts in the curtain of flame, stood a great granite monolith some eight feet in height; on top of which, incongruous with its diminutiveness, rested the noxious carven statuette. From a wide circle of ten scaffolds set up at regular intervals with the flame-girt monolith as a centre hung, head downward, the oddly marred bodies of the helpless squatters who had disappeared. It was inside this circle that the ring of worshippers jumped and roared, the general direction of the mass motion being from left to right in endless Bacchanal between the ring of bodies and the ring of fire."
...but I guess it would be quite disturbing to your neighbours ;-)
The altar is more for us, than for The Great Old Ones. It directs our mind towards Them, it helps us to focus. So the altar should be in my opininon both personal and also focused, directed towards your goal. Is it place of worship, reverence? Or is it a place of sacrifice? Or perhaps one that is for magical work? A lot depends on purpose.
My own, due to spatial limitations is still under desingn, but definitely will be small. But I do feel one is needed, so one will be created. How do you feel about it? Listen to yourself in darkness and you should find an answer...
Man you are going to LOVE The Bible of Cthulhu
@Nyarlathoth I do hope so. I'm going to listen to some of the recorded sessions to get a feeling this weekend. But we digress...
WHAT IS THE CULT?