What are the proper forms that I need to have completed to make sure a SPC gets promoted to SGT in the NG?

I didnt see this stated by you, but make sure your soldier is there with you while you do it. Have them learn by your aide and they can also ensure they have all their points and such lines up/nothing is missing.

Start with what you know he needs for the promotion board:
promotion checklist from your state
NGB form 4100-1-A-R-E
current ERB
current individual weapon qual
current APFT and HT-WT (DA 705/5500 last 3 yrs)
all awards (approved 638s and orders)
all mil ed (DA 1059s and certs)
all civ ed (transcripts)
evaluations (explained below)
For E5, all of this is submitted as a packet FROM THE SOLDIER, through unit to BN.
You need to get the 4100 from your unit or BN S1.
Everything else except for the evals, the Soldier provides from his iPERMS or from his own pile at home of docs that he should have submitted for iPERMS.
The Specialist Evaluation Forms are where you can help. There should be one for every year in the unit (kinda like an NCOER), filled out by first line leader, and added to the promotion packet.
Those go along with everything listed above through your company to your BN S1. that needs to be submitted by mid-JAN. Your BN or BDE will conduct a decentralized E5 promotion board with paper packets around April, so that pdf file you submit should result in legible printouts for board members to read.

If you can get your hands on your packet, you could use that as a template.