What is the process of getting a divorce? If dual-military, does the military have to be involved?

JAG isn't even allowed to help you with your divorce. The only thing you need to know as far as the military is concerned is that you are not divorced until you're completely divorced. You could be separated for 5 years and living in separate states, but that doesn't mean you can start dating yet. North Carolina has a mandatory one year waiting period to file as well.
Don't forget to update your BAH status once it's over. You don't want to be on the hook for paying back BAH, or worse, investigated for BAH fraud.

Definitely, absolutely, retain a civilian lawyer, even if you think you can't afford it. You can't afford not to.

Make sure you have a lawyer who understands military entitlements so that becomes part of the divorce settlement since you have a child.

Find out what your rights are. Don't get burned in the process. You need to know what state and military laws and regulations relate to your separation and divorce.

1) Do not seek advice here.
2) Seek advice from a lawyer.

It's not the Army's business, other than changing your marital status, any name change, and maybe DEERS, once the divorce is final. Also, the parent with custody will likely need to update the Family Care Plan.
Even though it may be uncontested, but there is a need to ensure the decree addresses the child's support, health insurance, education (who pays what for college), visitation (who gets what holidays/birthdays and who pays transportation each way), and possibly a requirement for one or both parents to maintain life insurance.
You may want to consider an attorney who does no fault divorces, if available in NC, as should he. It may be amicable now, but if either or both of you remarry the friendliness may expire.

I would think that you'd just get legal representation and once the documentation is finalized you'd provide copies to your command so the proper adjustments can be made.

You should get a lawyer. In NC in order to be considered “seperated” you cannot live under the same roof. There is a 1 year separation requirement, it does not need to be proven, but it can be contested. You can inform your command as a FYI, but there is not a lot of involvement outside if family care plans and general information about where you live and if you can support yourself and dependant. Just be open and honest with your leaders, thats always the best route. As far as JAG is concerned, they offer a separation and divorce information brief I believe. Check out the Legal section in the Soldier Support Center (SSC first floor) for guidence. Good luck.

SPC (Join to see) Others mention getting an attorney. Great, just like taking your car to an independent repair shop, PLEASE NETWORK YOUR FRIENDS TO FIND A RESPECTABLE ATTORNEY WITH A GOOD REPUTATION !!!

I am sorry to hear about the fact you are getting a divorce. As a family law attorney, I strongly advise that you seek out an attorney who practices in the area of family law with an emphasis on military issues. There are several issues which immediately pop out at me from your question. These are way more than I can begin to address in this forum. However, I can tell you that I have several clients who thought they could do their divorce on their own or with the help of a paralegal. They end up having to hire me, or another attorney, because of so many screw up with the military aspects of the divorce that were either omitted or not thoroughly understood.
In North Carolina there are several attorneys who practice in the area of family law and have military issues. Mark Sullivan wrote the book on military divorce, The Military Divorce Handbook. He and his partners are excellent in dealing with your situation. His phone number is [login to see] . If he is not available, I know that he can recommend someone to represent you. Also, you can go to JAG and ask them to submit your information to the ABA Military Pro Bono Representation Project. Mary Meixner at the ABA handles these cases and finds attorneys in your state to represent you for free.
You do need to inform your command that you are going to get a divorce. There are several reasons to do this. If you are getting a divorce, the military command needs to know in case there are financial issues, family care issues, etc. You don't need command to find out from someone else.