How can an amateur introduce a cryptographic algorithm?
First, if it's a secret-key cipher, then you'll never find any proof of security—at best you'll, exhaust the standard cryptanalysis techniques and exhaust the brilliant ideas of the brilliant cryptanalysts.  If it's a public-key cryptosystem, then at best you can prove reduction to a widely understood uniformly hard problem.  If you're not already familiar with these facts, you won't get the attention of serious cryptanalysts.

Second, you need to convincingly demonstrate (a) that you understand the standard cryptanalysis techniques, and (b) that it does not fall to them.  Cryptanalysts don't want to waste time doing your homework.  If you haven't already done your homework, you won't get the attention of serious cryptanalysts.

There's no good textbook of which I'm aware on this subject, but you might start with


  Bruce Schneier, ‘Self-Study Course in Block Cipher Cryptanalysis’, Cryptologia, 24(1), January 2000, pp. 18–34.
  https://www.schneier.com/academic/archives/2000/01/self-study_course_in.html


Third, if you want this to be anything more than an amusing intellectual exercise for bored cryptanalysts whose attention you've gotten, your cryptosystem has to fill a niche that nothing else does.  Is it faster or lower-energy on some significant platform than every other cipher providing comparable security?  Does it fill a specific need that no other cipher does, like a weird block size that is important for some particular reason?

For example, Gimli is a sponge with a much smaller state than Keccak-f1600, conveniently fitting in registers on more platforms, out of which one can build all manner of secret-key crypto gizmos.  (Gimli is a very new design, with some preliminary theoretical cryptanalysis.)

If you propose a general-purpose stream cipher that's not faster than software ChaCha or hardware AES-CTR, or a general-purpose collision-resistant hash function that's not faster than software BLAKE2b or hardware SHA-3, you won't get the attention of serious cryptanalysts who aren't either just bored or hoping their novel pet cryptanalysis techniques that failed to break the aforementioned ones might break something.

This applies even to designs that already have been published: Maybe RIPEMD160 resists collisions better than SHA-1 does, but it's not worth the time of most cryptanalysts to study it.  Maybe CAST6 is a better 128-bit block cipher than AES, but it's not worth the time of most cryptanalysts to study it.

Finally, if you want to get the attention of cryptanalysts, don't start with your cipher—start by showing that you are a competent cryptanlyst by using novel techniques to break crypto!  Best if you can break existing cryptosystems that were previously unbroken, or break them worse than the best published attacks.  This may be easier if you study cryptosystems that have received relatively little attention, including your own.

If you have a write-up demonstrating all of this—either a description of a novel cipher with evidence of resistance to standard cryptanalytic techniques and a reason to be interesting, or a novel cryptanalytic attack on an existing cipher—then a first step would be to submit it to the IACR cryptology eprint archive.  That won't impart the kind of serious review that, e.g., the AES and SHA-3 competitions did, of course, but it may catch the notice of some cryptologists and will be easier to show to someone whose review you are seeking, and if you do get the attention of a serious academic cryptologist they will likely have followup suggestions for publication venues.