What you really need is the name of the unit. The Battalion/Division you belonged to.
Once you have that, you would be able to request the record of trial. The record of trial is the official report of the court martial
You can request the record of trial under the Freedom Of Information Act (FOIA).
If the record still exists. But if its a attempted murder trial, I suspect there is still a record.
I would start with a request of your records. You can do so herehttp://www.archives.gov/veterans/military-service-records/standard-form-180.html
If you were in the same unit, you can then submit a FOIA request to that same battalion. I would also request to the Division.
The FOIA request must be in writing, state that its a request under the FOIA. You will want to request the record of trial for US v Neely. I would also request under the various other possible names. Your request must state your willing to pay the fees to copy the documents.
You send the request to the adjutant of the unit (battalion and division)
If the record exits, again I suspect it will, you can get a copy that way.
Let me know if you have more questions