What to do if I'm filing per se my own lawsuit in line with the arbitration code and I need to make sure I am doing so correctly?

I am curious about why you would attempt to prosecute a medical malpractice case by yourself and "have no interest in hiring an attorney."  Medical malpractice cases are complex and have specific requirements (such as expert testimony, statutes of limitations, special notices, limitations on damages, special proof for future damages, discovery rules, etc.). I cannot imagine a lay person handling this by himself or herself.

You ask about arbitration.  Is arbitration required in your case?  If so, I suspect you are in even worse shape without an attorney.  In my experience (which does not include the State of Washington), arbitration clauses and proceedings strongly (did I mention "strongly"?) favor the defendant.  It is difficult to get a fair opportunity with an attorney; it will be worse without one.  If your case is subject to arbitration because the Washington statutes require it (your claim is less than $15,000 or in some jurisdictions less than $50,000), you may have a better shot.  Even then, it will not be easy to obtain the necessary expert testimony and dispute the defendant's expert testimony.

If you are determined to proceed without an attorney (or cannot get an attorney to represent you), there are fewer technical requirements in arbitration than in court.  The form that says you must proceed in arbitration will tell you something about how to proceed.  It should say who will conduct the arbitration - the AAA (American Arbitration Association)?  Washington State Law?  Some professional organization?  Those organizations should have rules that tell you how to proceed.  If this is an arbitration required by Washington law, I suggest you ask the Clerk of Court where you can obtain assistance and/or contact the nearest law school to see if they can recommend a source to help you. 

Good luck.