What is the best way to obtain a position within USMA West Point?

Concur with LCDR Joshua Gillespie said below about what you re stepping into. We, as cadets and midshipmen need you senior enlisted advisors there. You fill a very big part.

As far as getting in, I would send your stuff to the CSM. I would also see if there are any other people you know that might either be there, or have been there. Ask them to write referral letters for you application. It is always better to have someone who has done well at a place to vouch for someone else.

I am not sure where you are at, but there is more than likely a West Point Alumni Association. Try and meet some of these people to see if they can help you. They generally know the ins and outs.

Best of luck!

I can't offer any wisdom on how the Academies select their staff; particularly Senior Enlisted...but I can offer some thoughts from the perspective of those you may ultimately have the opportunity to help mentor and prepare for careers as future officers. When I was a Mid, our SEAs were revered. I had a Navy SEAL (E-8) and Marine Corps Gunnery Sergeant who led my class through much of our initial training. To us, these people were supreme professionals who gave us our first real look at the services we were aspiring to join. The Academy training program is run, in no small part, by the "upper class", so it's very important "we" get the opposing perspective from the "Fleet"...or Army, in this case. Many of the folks "flaming" us may end up being very close to peers after graduation...but Senior Enlisted Advisors and Company Officers will always be mentors young JOs look up to as they start their careers. I hope you get the assignment, and envy you more than a little...Best of Luck!

USMA instructor is a nominative assignment, so the right way would be to contact either your branch PDNCO, branch chief, or the USMA commandant or CSM.
However, what you have going against you is that you would be going from TDA to TDA. It's prohibited by HRC policy and against your professional development model. That means it would put you behind your peers in the future. It would also require a waiver from HRC to do the move and you may be a lower choice candidate until you complete your KD time in your new grade.